arxiv:2009.04980v2 [math.lo] 14 feb 2021

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INFINITESIMAL ANALYSIS WITHOUT THE AXIOM

OF CHOICE

KAREL HRBACEK AND MIKHAIL G KATZ

Abstract It is often claimed that analysis with infinitesimalsrequires more substantial use of the Axiom of Choice than tradi-tional elementary analysis The claim is based on the observationthat the hyperreals entail the existence of nonprincipal ultrafil-ters over N a strong version of the Axiom of Choice while thereal numbers can be constructed in ZF The axiomatic approachto nonstandard methods refutes this objection We formulate atheory SPOT in the st-isin-language which suffices to carry outinfinitesimal arguments and prove that SPOT is a conservativeextension of ZF Thus the methods of Calculus with infinitesimalsare just as effective as those of traditional Calculus The conclu-sion extends to large parts of ordinary mathematics and beyondWe also develop a stronger axiomatic system SCOT conservativeover ZF + ADC which is suitable for handling such features asan infinitesimal approach to the Lebesgue measure Proofs of theconservativity results combine and extend the methods of forcingdeveloped by Enayat and Spector

Contents

1 Introduction 211 Axiom of Choice in Mathematics 212 Countering the objection 413 SPOT and SCOT 52 Theory SPOT and Calculus with infinitesimals 721 Some consequences of SPOT 722 Mathematics in SPOT 113 Theory SCOT and Lebesgue measure 134 Conservativity of SPOT over ZF 1741 Forcing according to Enayat and Spector 17

Date February 9 20212020 Mathematics Subject Classification Primary 26E35 Secondary 03A05 03C2503C62 03E70 03H05Key words and phrases nonstandard analysis axiom of choice ultrafilter forcingextended ultrapower

1

2 KAREL HRBACEK AND MIKHAIL G KATZ

42 Extended ultrapowers 215 Conservativity of SCOT over ZFc 246 Standardization for parameter-free formulas 287 Idealization 3171 Idealization over uncountable sets 3272 Further theories 348 Final Remarks 3681 Open problems 3682 Forcing with filters 3783 Zermelo set theory 3784 Weaker theories 3785 Finitistic proofs 3886 SPOT and CH 3887 External sets 399 Conclusion 40References 41

1 Introduction

Many branches of mathematics exploit the Axiom of Choice (AC)to one extent or another It is of considerable interest to gauge howmuch the Axiom of Choice can be weakened in the foundations ofnonstandard analysis Critics of analysis with infinitesimals often claimthat nonstandard methods require more substantial use of AC thantheir standard counterparts The goal of this paper is to refute such aclaim

11 Axiom of Choice in Mathematics We begin by consider-ing the extent to which AC is needed in traditional non-infinitesimalmathematics Simpson [35] introduces a useful distinction between set-theoretic mathematics and ordinary or non-set-theoretic mathematicsThe former includes such disciplines as general topology abstract al-gebra and functional analysis It is well known that fundamental the-orems in these areas require strong versions of AC Thus

bull Tychonoffrsquos Theorem in general topology is equivalent to fullAC (over Zermelo-Fraenkel set theory ZF)bull Prime Ideal Theorem asserts that every ring with unit has a

(two-sided) prime ideal PIT is an essential result in abstractalgebra and is ldquoalmostrdquo as strong as AC (it is equivalent overZF to Tychonoffrsquos Theorem for Hausdorff spaces) It is also

INFINITESIMAL ANALYSIS WITHOUT THE AXIOM OF CHOICE 3

equivalent to the Ultrafilter Theorem Every proper filter overa set S (ie in P(S)) can be extended to an ultrafilterbull Hahn-Banach Theorem for general vector spaces is equivalent to

the statement that every Boolean algebra admits a real-valuedmeasure a form of AC that is somewhat weaker than PIT

Jech [20] and Howard and Rubin [15] are comprehensive references forthe relationships between these and many other forms of AC

Researchers in set-theoretic mathematics have to accept strong formsof AC as legitimate whether or not they use nonstandard methods Ourconcern here is with ordinary mathematics which according to Simp-son includes fields such as the Calculus countable algebra differentialequations and real and complex analysis It is often felt that results inthese fields should be effective in the sense of not being dependent onAC However even these branches of mathematics cannot do entirelywithout AC There is a number of fundamental classical results thatrely on it they include

bull the equivalence of continuity and sequential continuity for real-valued functions on Rbull the equivalence of the ε-δ definition and the sequential definition

of closure points for subsets of Rbull closure of the collection of Borel sets under countable unions

and intersectionsbull countable additivity of Lebesgue measure

Without an appeal to AC one cannot even prove that R is not a unionof countably many countable sets or that a strictly positive functioncannot have vanishing Lebesgue integral (Kanovei and Katz [23]) How-ever these results follow already from ACC the Axiom of Choice forCountable collections a weak version of AC that many mathemati-cians use without even noticing1 Nevertheless it is true enough thatno choice is needed to define the real number system itself or to developthe Calculus and much of ordinary mathematics

It has to be emphasized that objections to AC are not a matterof ontology but of epistemology In other words the issue is not theexistence of objects but proof techniques and procedures For betteror worse many mathematicians nowadays believe that the objects ofinterest to them can be represented by set-theoretic structures in a uni-verse that satisfies ZFC Zermelo-Fraenkel set theory with the Axiomof Choice Nevertheless they may prefer results that are effective thatis do not use AC For the purposes of this discussion mathematical

1For example Halmos [10] p 42 see [4] Sec 57 for further discussion


results are effective if they can be proved in ZF Much of ordinarymathematics is effective in this sense

We now consider whether nonstandard methods require anythingmore A common objection to infinitesimal methods in the Calculusis the claim that the mere existence of the hyperreals2 implies theexistence of a nonprincipal ultrafilter U over N The proof is simpleFix an infinitely large integer ν in lowastN N and define U sube P(N) byX isin U larrrarr ν isin lowastX forX sube N It is easy to see that U is a nonprincipalultrafilter over N For example if X cup Y isin U then ν isin lowast(X cup Y ) =lowastX cup lowastY where the last step is by the Transfer Principle Hence eitherν isin lowastX or ν isin lowastY and so X isin U or Y isin U If X is finite then X = lowastX hence ν isin lowastX and so U is nonprincipal

By the well-known result of Sierpinski [34] (see also Jech [20] Prob-lem 110) U is a non-Lebesgue-measurable set (when subsets of N areidentified with real numbers in some natural way) In the celebratedmodel of Solovay [36] ZF +ACC holds (even the stronger ADC theAxiom of Dependent Choice holds there) but all sets of real numbersare Lebesgue measurable hence there are no nonprincipal ultrafiltersover N in this model The existence of nonprincipal ultrafilters over Nrequires a strong version of AC such as PIT it cannot be proved inZF (or even ZF + ADC)

12 Countering the objection How can such an objection be an-swered As in the case of the traditional mathematics the key is to looknot at the objects but at the methods used Currently there are twopopular ways to practice Robinsonrsquos nonstandard analysis the model-theoretic approach and the axiomaticsyntactic approach Analysiswith infinitesimals does not have to be based on hyperreal structuresin the universe of ZFC It can be developed axiomatically the mono-graph by Kanovei and Reeken [24] is a comprehensive reference for suchapproaches Internal axiomatic presentations of nonstandard analysissuch as IST or BST extend the usual isin-language of set theory bya unary predicate st (st(x) reads x is standard) For reference theaxioms of BST are stated in Section 7

It is of course possible to weaken ZFC to ZF within BST or ISTbut this move by itself does not answer the above objection It is easilyseen by a modification of the argument given above for hyperreals thatthe theory obtained from BST or IST by replacing ZFC by ZF proves

2By the hyperreals we mean a proper elementary extension of the reals ie a properextension that satisfies Transfer The definite article is used merely for grammaticalcorrectness Subsets of N can be identified with real numbers see SP rArr SP

prime inthe proof of Lemma 24 for one way to do that


PIT (Hrbacek [16]) This argument uses the full strength of the prin-ciples of Idealization and Standardization (see Section 7) HoweverCalculus with infinitesimals can be fully carried out assuming muchless Examination of texts such as Keisler [26] and Stroyan [40] revealsthat only very weak versions of these principles are ever used there Ofcourse one has to postulate that infinitesimals exist (Nontriviality) butstronger consequences of Idealization are not needed As for Standard-ization these textbooks only explicitly postulate a special consequenceof it namely the following principle

SP (Standard Part) Every limited real is infinitely close to a stan-dard real

see Keisler [26 27] Axioms A - E However this is somewhat mis-leading Keisler does not develop the Calculus from his axioms alonethey describe some properties of the hyperreals but the hyperreals areconsidered to be an extension of the field R of real numbers in theuniverse of ZFC and the principles of ZFC can be freely used Inparticular the principle of Standardization is not an issue it is auto-matically satisfied for any formula While Standardization for formulasabout integers appears innocuous Standardization for formulas aboutreals can lead to the existence of nonprincipal ultrafilters On the otherhand some instances of Standardization over the reals are unavoidablefor example to prove the existence of the function f prime (the derivative off) defined in terms of infinitesimals for a given real-valued function fon R

13 SPOT and SCOT In the present text we introduce a theorySPOT in the st-isin-language a subtheory of IST and BST and weshow that SPOT proves Countable Idealization and enough Standard-ization for the purposes of the Calculus We use forall and exist as quantifiersover sets and forallst and existst as quantifiers over standard sets The axiomsof SPOT are

ZF (Zermelo - Fraenkel Set Theory)

T (Transfer) Let φ be an isin-formula with standard parameters Then

forallstx φ(x)rarr forallx φ(x)

O (Nontriviality) existν isin N forallstn isin N (n 6= ν)

SPprime (Standard Part)

forallA sube N existstB sube N forallstn isin N (n isin B larrrarr n isin A)

Our main result is the following


Theorem A The theory SPOT is a conservative extension of ZF

Thus the methods used in the Calculus with infinitesimals do notrequire any appeal to the Axiom of Choice

The result allows significant strengthenings We let SN be the Stan-dardization principle for st-isin-formulas with no parameters (see Sec-tion 6) The principle allows Standardization of much more complexformulas than SPOT alone

Theorem B The theory SPOT + SN is a conservative extensionof ZF

It is also possible to add some Idealization We let BIprime be BoundedIdealization (see Section 7) for isin-formulas with standard parameters

Theorem C The theory SPOT+B+BIprime is a conservative exten-sion of ZF

This is the theory BST with ZFC replaced by ZF Standardizationweakened to SP and Bounded Idealization weakened to BIprime we denoteit BSPTprime This theory enables the applicability of some infinitesimaltechniques to arbitrary topological spaces It also proves that there isa finite set S containing all standard reals a frequently used idea

As noted above some important results in elementary analysis andelsewhere in ordinary mathematics require the Axiom of CountableChoice On the other hand ACC entails no ldquoparadoxicalrdquo conse-quences such as the existence of Lebesgue-non-measurable sets or theexistence of an additive function on R different from fa x 7rarr ax for alla isin R Many mathematicians find ACC acceptable These considera-tions apply as well to the following stronger axiom

ADC (Axiom of Dependent Choice) If R is a binary relation on aset A such that foralla isin A existaprime isin A (aRaprime) then for every a isin A thereexists a sequence 〈an | n isin N〉 such that a0 = a and anRan+1 for alln isin N

This axiom is needed for example to prove the equivalence of the twodefinitions of a well-ordering (Jech [21] Lemma 52)

(1) Every nonempty subset of a linearly ordered set (Alt) has aleast element

(2) A has no infinite decreasing sequence a0 gt a1 gt gt an gt

We denote by ZFc (ldquoZFC Literdquo) the theory ZF + ADC Thistheory is sufficient for axiomatizing ordinary mathematics (and many


SPOT SPOT+SN SCOT BSPTprime BSCTprime BST

isin-theory ZF ZF ZF+ADC ZF ZF+ADC ZFCTransfer yes yes yes yes yes yesIdealization countable countable countable standard

paramsstandardparams

full

Standardiz SP SPstandardparams

SCstandardparams

SP SC full

Countablest-isinChoice

no no yes no yes Standard-sizeChoice

Table 1 Theories and their properties

results of set-theoretic mathematics as well) Let SCOT be the theoryobtained from SPOT by strengthening ZF to ZFc SP to Countablest-isin-Choice (CC) and adding SN see Section 3

Theorem D The theory SCOT is a conservative extension of ZFc

In SCOT one can carry out most techniques used in infinitesimaltreatments of ordinary mathematics As examples we give a proofof Peanorsquos Existence Theorem and an infinitesimal construction ofLebesgue measure in Section 3 Thus the nonstandard methods used inordinary mathematics do not require any more choice than is generallyaccepted in traditional ordinary mathematics

Further related conservative extension theorems can be found in Sec-tions 5 6 and 7

2 Theory SPOT and Calculus with infinitesimals

21 Some consequences of SPOT The axioms of SPOT weregiven in Section 13

Lemma 21 The theory SPOT proves the following

forallstn isin N forallm isin N (m lt nrarr st(m))

Proof Given a standard n isin N and m lt n let A = k isin N | k lt mBy SPprime there is a standard B sube N such that for all standard k k isin Biff k isin A iff k lt m The set B sube N is bounded above by n (Transfer)so it has a greatest element k0 (lt is a well-ordering of N) which


is standard by Transfer Now we have k0 lt m and k0 + 1 ≮ m sok0 + 1 = m and m is standard

Lemma 22 (Countable Idealization) Let φ be an isin-formula with ar-bitrary parameters The theory SPOT proves the following

forallstn isin N existx forallm isin N (m le n rarr φ(m x))larrrarr existx forallstn isin N φ(n x)

Proof If forallstn isin N φ(n x) then for every standard n isin N forallm isinN (m le nrarr φ(m x)) by Lemma 21

Conversely assume forallstn isin N existx forallm isin N (m le n rarr φ(m x)) Bythe Axiom of Separation of ZF there is a set

S = n isin N | existx forallm isin N (m le nrarr φ(m x))

and the assumption implies that forallstn isin N (n isin S)Assume that S contains standard integers only Then N S 6= empty by

the axiom O Let ν be the least element of NS Then ν is nonstandardbut ν minus 1 is standard a contradiction

Let micro be some nonstandard element of S We have existx forallm isin N (m lemicro rarr φ(m x)) as n le micro holds for all standard n isin N we obtainexistx forallstn isin N φ(n x)

Countable Idealization easily implies the following more familiarform We use forallst fin and existst fin as quantifiers over standard finite sets

Corollary 23 Let φ be an isin-formula with arbitrary parameters Thetheory SPOT proves the following For every standard countable set A

forallstfina sube A existx forally isin a φ(x y)larrrarr existx forallsty isin A φ(x y)

The axiom SPprime is often stated and used in the form

(SP) forallx isin R (x limited rarr existstr isin R (x asymp r))

where x is limited iff |x| le n for some standard n isin N and x asymp r iff|xminus r| le 1n for all standard n isin N n 6= 0 The unique standard realnumber r is called the standard part of x or the shadow of x notationsh(x)

We note that in the statement of SPprime N can be replaced by anycountable standard set A

Lemma 24 The statements SPprime and SP are equivalent (over ZF +O + T)

Proof of Lemma 24 SPprime rArr SP Assume x isin R is limited by a stan-dard n0 isin N Let A = q isin Q | q le x Applying SPprime with N replacedby Q we obtain a standard set B sube Q such that forallstq isin Q (q isin B larrrarrq isin A) As forallstq isin B (q le n0) holds the set B is bounded above (apply


Transfer to the formula q isin B rarr q le n0) and so it has a supremumr isin R which is standard (Transfer again) We claim that x asymp r Ifnot then |x minus r| gt 1

nfor some standard n hence either x lt r minus 1

n

or x gt r + 1n In the first case supB le r minus 1

nand in the second

supB ge r + 1n either way contradicts supB = r

SPrArr SPprime The obvious idea is to represent the characteristic func-tion of a set A sube N by the binary expansion of a real number in [0 1]But some real numbers have two binary expansions and therefore cor-respond to two distinct subsets of N This is a source of technicalcomplications that we avoid by using decimal expansions instead

Given A sube N let χA be the characteristic function of A Define a

real number xA = Σinfinn=0

χ(n)10n+1 as 0 le xA le

19 there is a standard real

number r asymp xA Let r = Σinfinn=0

an10n+1 be the decimal expansion of r

where for every n there is k gt n such that ak 6= 9 Note that if n isstandard then there is a standard k with this property by Transfer Ifχ(n) = an for all n then A is standard and we let B = A Otherwiselet n0 be the least n where χ(n) 6= an From r asymp xA it follows easilythat n0 is nonstandard In particular an isin 0 1 holds for all standardn hence by Transfer for all n isin N Let B = n isin N | an = 1 ThenB is standard and for all standard n isin N n isin B iff an = 1 iff χ(n) = 1iff n isin A

As explained in the Introduction Standardization over uncountablesets such as R even for very simple formulas implies the existence ofnonprincipal ultrafilters over N and so it cannot be proved in SPOT

(consider a standard set U such that forallstX (X isin U larrrarr X sube N and ν isinX) where ν is a nonstandard integer) But we need to be able toprove the existence of various subsets of R and functions from R to Rthat arise in the Calculus and may be defined in terms of infinitesimalsUnlike the undesirable example above such uses generally involve Stan-dardization for formulas with standard parameters

An st-isin-formula Φ(v1 vn) is ∆st if it is of the form

Qst

1 x1 Qst

m xm ψ(x1 xm v1 vn)

where ψ is an isin-formula and Q stands for exist or forall

Lemma 25 Let Φ(v1 vn) be a ∆st formula with standard param-eters Then SPOT proves forallstS existstP forallstv1 vn(

〈v1 vn〉 isin P larrrarr 〈v1 vn〉 isin S and Φ(v1 vn))

Proof Let Φ(v1 vn) beQst

1 x1 Qst

m xm ψ(x1 xm v1 vn) andφ(v1 vn) be Q1 x1 Qm xm ψ(x1 xm v1 vn) By TransferΦ(v1 vn) larrrarr φ(v1 vn) for all standard v1 vn The set P =


〈v1 vn〉 isin S | φ(v1 vn) exists by the Separation Principle ofZF and has the required property

This result has twofold importance

bull The meaning of every predicate that for standard inputs is de-fined by a Qst

1 x1 Qst

m xm ψ formula with standard parametersis automatically extended to all inputs where it it given by theisin-formula Q1 x1 Qm xm ψbull Standardization holds for all isin-formulas with additional predi-

cate symbols as long as all these additional predicates are de-fined by ∆st formulas with standard parameters

In BST all st-isin-formulas are equivalent to ∆st formulas (see Kanoveiand Reeken [24] Theorem 323) In SPOT the equivalence is true onlyfor certain classes of formulas but they include definitions of all thebasic concepts of the Calculus and much beyond

We recall that h isin R is infinitesimal iff 0 lt |h| lt 1n

holds for all

standard n isin N n gt 0 We use forallin and existin for quantifiers ranging overinfinitesimals and 0 The basic concepts of the Calculus have infini-tesimal definitions that involve a single alternation of such quantifiersThe following proposition strengthens a result in Vopenka [41] p 148It shows that the usual infinitesimal definitions of Calculus conceptsare ∆st The variables x y range over R and mn ℓ range over N0

Proposition 26 In SPOT the following is true Let φ(x y) be anisin-formula with arbitrary parameters Then forallinh existink φ(h k)larrrarr

forallstm existstn forallx [ |x| lt 1nrarr existy (|y| lt 1m and φ(x y)) ]

By duality we also have existinh forallink φ(h k)larrrarr

existstm forallstn existx [ |x| lt 1n and forally (|y| lt 1mrarr φ(x y)) ]

Proof The formula forallinh existink φ(h k) means

forallx [ forallstn (|x| lt 1n)rarr existy forallstm (|y| lt 1m and φ(x y)) ]

where we assume that the variables mn do not occur freely in φ(x y)Using Countable Idealization (Lemma 22) we rewrite this as

forallx [ forallstn (|x| lt 1n)rarr forallstm existy forallℓ le m (|y| lt 1ℓ and φ(x y)) ]

We now use the observation that forallℓ le m (|y| lt 1ℓ) is equivalentto |y| lt 1m and the rules (α rarr forallstv β) larrrarr forallstv (α rarr β) and(forallstv β rarr α)larrrarr existstv (β rarr α) valid assuming that v is not free in α(note that Transfer implies existn st(n)) This enables us to rewrite thepreceding formula as follows

forallx forallstm existstn [ |x| lt 1nrarr existy (|y| lt 1m and φ(x y)) ]


After exchanging the order of the first two universal quantifiers weobtain the formula

forallstm forallx existstn [ |x| lt 1nrarr existy |y| lt 1m and φ(x y)) ]

to which we apply (the dual form of) Countable Idealization to get

forallstm existstn forallx existℓ le n [ |x| lt 1ℓrarr existy (|y| lt 1m and φ(x y)) ]

After rewriting existℓ le n [ |x| lt 1ℓrarr ] as [forallℓ le n ( |x| lt 1ℓ)rarr ]and replacing forallℓ le n (|x| lt 1ℓ) by |x| lt 1n we obtain


proving the proposition

22 Mathematics in SPOT We give some examples to illustratehow infinitesimal analysis works in SPOT

Example 27 If F is a standard real-valued function on an openinterval (a b) in R and a b c d are standard real numbers with c isin(a b) we can define

(1) F prime(c) = dlarrrarr forallinh existink

(h 6= 0rarr

F (c+ h)minus F (c)

h= d+ k

)

Let Φ(F c d) be the formula on the right side of the equivalence in (1)Lemma 25 establishes that the formula Φ is equivalent to a ∆st for-mula and φ(F c d) provided by the proof of Lemma 25 is easily seento be equivalent to the standard ε-δ definition of derivative For anystandard F the set F prime = 〈c d〉 | φ(F c d) is standard it is thederivative function of F

Proposition 26 generalizes straightforwardly to all formulas thathave the form Ah1 Ahn Ek1 Ekm φ(h1 k1 v1 ) orEh1 Ehn Ak1 Akm φ(h1 k1 v1 ) where each A is eitherforall or forallin and each E is either exist or existin All such formulas are equivalentto ∆st formulas

Formulas of the form Q1h1 Qnhn φ(h1 hn v1 vk) whereeach Q is either forall or forallin or exist or existin but all quantifiers over infinitesimalsare of the same kind (all existential or all universal) are also ∆st Asan example existinh forally existink φ(h k x y) is equivalent to

existh forally existk forallstm forallstn (|h| lt 1m and |k| lt 1n and φ(h k x y))

The two quantifiers over standard elements of N can be replaced by asingle one

existh forally existk forallstm (|h| lt 1m and |k| lt 1m and φ(h k x y))

and then moved to the front using Countable Idealization


Klein and Fraenkel proposed two benchmarks for a useful theory ofinfinitesimals (see Kanovei et al [22])

bull a proof of the Mean Value Theorem by infinitesimal techniquesbull a definition of the definite integral in terms of infinitesimals

The theory SPOT easily meets these criteria The usual nonstandardproof of the Mean Value Theorem (Robinson [31] Keisler [26 27]) usesStandard Part and Transfer and is easily carried out in SPOT Thefamiliar infinitesimal definition of the Riemann integral for standardbounded functions on a standard interval [a b] also makes sense inSPOT and can be expressed by a ∆st formula In the next examplewe outline a treatment inspired by Keislerrsquos use of hyperfinite Riemannsums in [27]

Example 28 Riemann Integral

We fix a positive infinitesimal h and the corresponding ldquohyperfinitetime linerdquo T = ti | i isin Z where ti = i middot h Let f be a standard real-valued function continuous on the standard interval [a b] Let ia ib besuch that ia middot hminus h lt a le ia middot h and ib middot h lt b le ib middot h + h Then

(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)

It is easy to show that the value of the integral does not depend on the

choice of h We thus have for standard f a b r int b

af(t) dt = r iff

forallinh existink(Σib

i=iaf(ti) middot h = r + k

)iff existinh existink

(Σib


)

The formulas are of the form AE and EE respectively and thereforeequivalent to ∆st formulas

The approach generalizes easily to the Riemann integral of boundedfunctions on [a b] We say that Th = 〈tprimei〉

ibi=ia

is an h- tagging on [a b] ifi middot h le tprimei le (i+ 1) middot h for all i = ia ibminus 1 and ib middot h le tprimeib le b Thenfor standard f a b r

bull f is Riemann integrable on [a b] andint b

af(x) dx = r iff

bull forallinh forallTh existink

(Σib

i=iaf(tprimei) middot h = r + k

)iff

bull existinh forallTh existink

(Σib


)

These formulas are again equivalent to ∆st formulas (the first one is ofthe form AAE and in the second one both quantifiers over standardsets are existential)

The tools available in SPOT enable nonstandard definitions andproofs in parts of mathematics that go well beyond the Calculus


Example 29 Frechet Derivative Given standard normed vectorspaces V and W a standard open subset U of V a standard functionf U rarr W a standard bounded linear operator A V rarr W and astandard x isin U A is the Frechet derivative of f at x isin U iff

forallz forallinh existink

( z V = h gt 0rarr

f(x + z)minus f(x)minus A middot z W z V

= k

)

This definition is equivalent to a ∆st formula

In Section 6 we show that Standardization for arbitrary formulaswith standard parameters can be added to SPOT and the resultingtheory is still conservative over ZF This result enables one to disposeof any concerns about the form of the defining formula

3 Theory SCOT and Lebesgue measure

We recall (see Section 13) that SCOT is SPOT + ADC + SN +CC where the principle CC of Countable st-isin-Choice postulates thefollowing

CC Let φ(u v) be an st-isin-formula with arbitrary parameters Thenforallstn isin N existx φ(n x)rarr existf (f is a function and forallstn isin Nφ(n f(n))

The set N can be replaced by any standard countable set A Weconsider also the principle SC of Countable Standardization

SC (Countable Standardization) Let ψ(v) be an st-isin-formula witharbitrary parameters Then

existstS forallstn (n isin S larrrarr n isin N and ψ(n))

Lemma 31 The theory SPOT + CC proves SC

Proof Let φ(n x) be the formula ldquo(ψ(n) and x = 0) or (notψ(n) and x = 1)rdquoIf f is a function provided by CC let A = n isin N | f(n) = 0 BySP there is a standard set S such that for all standard n isin N n isin Siff n isin A iff ψ(n) holds

We introduce an additional principle CCst

CCst Let φ(u v) be an st-isin-formula with arbitrary parametersThen

forallstn isin N existstxφ(n x)rarr existstF (F is a function and forallstn isin Nφ(n F (n))

The principle CCst

R is obtained from CCst by restricting the rangeof the variable x to R

Lemma 32 The theory SPOT + CC proves CCst

R


Proof First use the principle CC to obtain a function f N rarr Rsuch that forallstn isin N (f(n) isin R and st(f(n)) and φ(n f(n)) Next define arelation r sube N times N by 〈nm〉 isin r iff m isin f(n) By Lemma 31 SCholds By SC there is a standard R sube N times N such that 〈nm〉 isin Riff 〈nm〉 isin r holds for all standard 〈nm〉 Now define F Nrarr R byF (n) = m | 〈nm〉 isin R The function F is standard and for everystandard n the sets F (n) and f(n) have the same standard elementsAs they are both standard it follows by Transfer that F (n) = f(n)

The full principle CCst can conservatively be added to SCOT seeProposition 56

A useful consequence of SC is the ability to carry out external in-duction

Lemma 33 (External Induction) Let φ(v) be an st-isin-formula witharbitrary parameters Then SPOT + SC proves the following

[φ(0) and forallstn isin N (φ(n)rarr φ(n+ 1))rarr forallstnφ(n) ]

Proof SC yields a standard set S sube N such that forallstn isin N (n isin S larrrarrφ(n)) We have 0 isin S and forallstn isin N (n isin S rarr n + 1 isin S) Thenforalln isin N (n isin S rarr n + 1 isin S) by Transfer and S = N by inductionHence forallstn isin Nφ(n) holds

In Example 36 it is convenient to use the language of external collec-tions Let φ(v) be an st-isin-formula with arbitrary parameters We use

dashed curly braces to denote the external collection x isin A | φ(x)

We emphasize that this is merely a matter of convenience writingz isin x isin A | φ(x) is just another notation for φ(z)

Standardization in BST implies the existence of a standard set Ssuch that forallstz (z isin S larrrarr z isin x isin A | φ(x) ) We do not have

Standardization over uncountable sets in SCOT but one importantcase can be proved

Lemma 34 Let φ(v) be an st-isin-formula with arbitrary parameters

Then SCOT proves that inf st r isin R | φ(r) exists

The notation indicates the greatest standard s isin R such that s le rfor all standard r with the property φ(r) (+infin if there is no such r)

Proof Consider S = q isin Q | existstr isin R (q ge r and φ(r)) As Q is

countable the principle SC implies that there is a standard set S suchthat forallq isin Q (q isin S larrrarr q isin S) Therefore inf S exists (+infin if S = empty)

and it is what is meant above by inf st r isin R | φ(r)

We give two examples of mathematics in SCOT


Example 35 Peano Existence Theorem Peanorsquos Theorem as-serts that every first-order differential equation of the form yprime = f(x y)has a solution (not necessarily a unique one) satisfying the initial con-dition y(0) = 0 under the assumption that f is continuous in a neigh-borhood of 〈0 0〉 The infinitesimal proof begins by constructing thesequences

x0 = 0 xk+1 = xk + h where h gt 0 is infinitesimal

y0 = 0 yk+1 = yk + h middot f(xk yk)

One then shows that there is N isin N such that xk yk are definedfor all k le N a = st(xN) gt 0 and for some standard M gt 0|yk| le M middot a holds for all k le N The desired solution is a stan-dard function Y [0 a] rarr R such that for all standard x isin [0 a] ifx asymp xk then Y (x) asymp yk On the face of it one needs Standardizationover R to obtain this function but in fact SC suffices Consider thecountable set A = (Q times Q) cap ([0 a] times [minusM middot aM middot a]) By SC thereis a standard Z sube A such that for all standard 〈x y〉〈x y〉 isin Z iffexistk le N (x asymp xk and (y asymp yk or y ge yk) Define a standard function Y0on Q cap [0 a] by Y0(x) = infy | 〈x y〉 isin Z It is easy to verify thatY0 is continuous on Q cap [0 a] and that its extension Y to a continuousfunction on [0 a] is the desired solution

Example 36 Lebesgue measure In a seminal paper [29] Loebintroduced measures on the external power set of lowastR which becameknown as Loeb measures and used them to construct the Lebesguemeasure on R Substantial use of external collections is outside thescope of this paper (see Subsection 87) but it is possible to eliminatethe intermediate step and give an infinitesimal definition a la Loeb ofthe Lebesgue measure in internal set theory We outline here how toconstruct the Lebesgue outer measure on R in SCOT

Let T be a hyperfinite time line (see Example 28) and let E sube R bestandard A finite set A sube T covers E if

forallt isin T (existstx isin E (t asymp x)rarr t isin A)

We define micromicromicro by setting

(3) micromicromicro(E) = inf st r isin R | r asymp |A| middot h for some A that covers E

The collection whose infimum needs to be taken is external but theexistence of the infimum is justified by Lemma 34 It is easy to seethat the value of micromicromicro(E) is independent of the choice of the infinitesimalh in the definition of T Thus the external function micromicromicro can be defined forstandard E sube P(R) by an st-isin-formula with no parameters (preface


the formula on the right side of (3) by forallinh or existinh) The principleSN (see Subsection 13 and Section 6) yields a standard function m onP(R) such that m(E) = micromicromicro(E) for all standard E sube R We prove thatm is σ-subadditive

Let E =⋃infin

n=0En where E and the sequence 〈En | n isin N〉 arestandard If Σinfin

n=0m(En) = +infin the claim is trivial so we assumethat m(En) = rn isin R for all n Fix a standard ε gt 0 For everystandard n isin N there exists A such that φ(nA) ldquoA covers En and|A| middot h lt rn + ε2n+1rdquo holds By Countable st-isin-Choice there is asequence 〈An | n isin N〉 such that for all standard n φ(nAn) holds ByCountable Idealization (ldquoOverspillrdquo) there is a nonstandard ν isin N suchthat |An| middot h lt rn + ε2n+1 holds for all n le ν We let A =

⋃ν

n=0AnClearly A is finite and covers E Thus for r = sh(|A| middot h) we obtainm(E) le r and

|A| middot h le Σνn=0|An| middot h lt Σν

n=0rn + ε

Since the sequence Σinfinn=0rn converges we have sh(Σν

n=0rn) = Σinfinn=0rn

and m(A) le Σinfinn=0rn + ε As this is true for all standard ε gt 0 we

conclude that m(E) le Σinfinn=0rn = Σinfin

n=0m(En)

For closed intervals [a b] m([a b]) = b minus a Compactness of [a b]implies that forallt isin T cap [a b] existstx isin E (t asymp x) Thus if A covers [a b]then A supe Tcap [a b] and for A = Tcap [a b] one sees easily that |A| middot h asymp(bminus a) With more work one can show that m(E) coincides with theconventionally defined Lebesgue outer measure of E for all standardE sube R See Hrbacek [17] Section 3 for more details and other equivalentnonstandard definitions of the Lebesgue outer measure3 One can defineLebesgue measurable sets from m in the usual way One can also defineLebesgue inner measure for standard E by

microminus(E) =supst r isin R | r asymp |A| middot h for some A such that

forallt isin T (t isin Ararr existstx isin E (t asymp x))

and prove that a standard bounded E sube R is Lebesgue measurable iffm(E) = mminus(E) and the common value is the Lebesgue measure of Esee Hrbacek [18]

3In [17] Remark (3) on page 22 it is erroneously claimed that the statement m1(A) =r is equivalent to an internal formula The existence of the function m1 there followsfrom Standardization just as in the case of m above


4 Conservativity of SPOT over ZF

In this section we apply forcing techniques to prove conservativity ofSPOT over ZF

Theorem 41 The theory SPOT is a conservative extension of ZFIf θ is an isin-sentence then (SPOT ⊢ θ ) implies that (ZF ⊢ θ )

Theorem 41 = Theorem A is an immediate consequence of the fol-lowing proposition

Proposition 42 Every countable model M = (MisinM) of ZF has acountable extension M

lowast = (Mlowastisinlowast st) to a model of SPOT in whichM is the class of all standard sets

Proof of Theorem 41 Suppose SPOT ⊢ θ but ZF 0 θ where θis an isin-sentence Then the theory ZF + notθ is consistent therefore ithas a countable model M by Godelrsquos Completeness Theorem UsingProposition 42 one obtains its extension M

lowast SPOT so in particularM

lowast θ and by Transfer in Mlowast M θ This is a contradiction

The rest of this section is devoted to the proof of Proposition 42

41 Forcing according to Enayat and Spector We combine theforcing notion used by Enayat [8] to construct end extensions of modelsof arithmetic with the one used by Spector in [38] to produce extendedultrapowers of models M of ZF by an ultrafilter U isinM

In this subsection we work in ZF define our forcing notion and proveits basic properties The next subsection deals with generic extensionsof countable models of ZF and the resulting extended ultrapowersThe general reference to forcing and generic models in set theory isJech [21]

The set of all natural numbers is denoted N and letters mn k ℓ arereserved for variables ranging over N The index set over which theultrapowers will eventually be constructed is denoted I In this sectionwe assume I = N A subset p of N is called unbounded if forallm existn isinp (n ge m) and bounded if it is not unbounded Of course unboundedis the same as infinite and bounded is the same as finite We usethis terminology with a view to Section 7 where the construction isgeneralized to I = Pfin(A) for any infinite set A The notation forallaai isin p(for almost all i isin p) means foralli isin p c for some bounded c

As usual the symbol V denotes the universe of all sets and Vα (αranges over ordinals) are the ranks of the von Neumann cumulativehierarchy We let F be the class of all functions with domain I Thenotation emptyk stands for the k-tuple 〈empty empty〉


Definition 43 Let P = p sube I | p is unbounded For p pprime isin P wesay that pprime extends p (notation pprime le p) iff pprime sube p

Let Q = q isin F | existk isin N foralli isin I (q(i) sube Vk and q(i) 6= empty) Thenumber k is the rank of q We note that q(i) for each i isin I and qitself are sets but Q is a proper class We let 1 = q where q(i) = emptyfor all i isin I 1 is the only q isin Q of rank 0

The forcing notion H is defined as follows H = PtimesQ and 〈pprime qprime〉 isin Hextends 〈p q〉 isin H (notation 〈pprime qprime〉 le 〈p q〉) iff pprime extends p rank qprime =kprime ge k = rank q and for almost all i isin pprime and all 〈x0 xkprimeminus1〉 isin q

prime(i)〈x0 xkminus1〉 isin q(i) Every 〈p q〉 isin H extends 〈p 1〉

The poset P is used to force a generic filter over I as in Enayat [8]and H forces an extended ultrapower of V by the generic filter U forcedby P It is a modification of the forcing notion from Spector [38] withthe difference that in [38] U is not forced but assumed to be a givenultrafilter in V

A set D sube P is dense in P if for every p isin P there is pprime isin Dsuch that pprime extends p We note that for any set S sube I the setDS = p isin P | p sube S or p sube I S is dense in P

Similarly a class E sube H is dense in H if for every 〈p q〉 isin H thereis 〈pprime qprime〉 isin E such that 〈pprime qprime〉 le 〈p q〉

The forcing language L has a constant symbol z for every z isin V(which we identify with z when no confusion threatens) and a constant

symbol Gn for each n isin N Given an isin-formula φ(w1 wr v1 vs)we define the forcing relation 〈p q〉 13 φ(z1 zr Gn1

Gns) for

〈p q〉 isin H by meta-induction on the logical complexity of φ We usenotand and exist as primitives and consider the other logical connectivesand quantifiers as defined in terms of these Usually we suppress theexplicit listing in φ of the constant symbols z for the elements of V

Definition 44 (Forcing relation)

(1) 〈p q〉 13 z1 = z2 iff z1 = z2(2) 〈p q〉 13 z1 isin z2 iff z1 isin z2(3) 〈p q〉 13 Gn1

= Gn2iff rank q = k gt n1 n2 and

forallaai isin p forall〈x0 xkminus1〉 isin q(i) (xn1= xn2

)

(4) 〈p q〉 13 Gn1isin Gn2

iff rank q = k gt n1 n2 andforallaai isin p forall〈x0 xkminus1〉 isin q(i) (xn1

isin xn2)

(5) 〈p q〉 13 Gn = z iff 〈p q〉 13 z = Gn iff rank q = k gt n andforallaai isin p forall〈x0 xkminus1〉 isin q(i) (xn = z)

(6) 〈p q〉 13 z isin Gn iff rank q = k gt n andforallaai isin p forall〈x0 xkminus1〉 isin q(i) (z isin xn)


(7) 〈p q〉 13 Gn isin z iff rank q = k gt n andforallaai isin p forall〈x0 xkminus1〉 isin q(i) (xn isin z)

(8) 〈p q〉 13 notφ(Gn1 Gns

) iff rank q = k gt n1 ns and there

is no 〈pprime qprime〉 extending 〈p q〉 such that 〈pprime qprime〉 13 φ(Gn1 Gns

)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff

〈p q〉 13 φ(Gn1 Gns

) and 〈p q〉 13 ψ(Gn1 Gns

)(10) 〈p q〉 13 existv ψ(Gn1

Gns v) iff rank q = k gt n1 ns and

for every 〈pprime qprime〉 extending 〈p q〉 there exist 〈pprimeprime qprimeprime〉 extending

〈pprime qprime〉 and m isin N such that 〈pprimeprime qprimeprime〉 13 ψ(Gn1 Gns

Gm)

Lemma 45 (Basic properties of forcing)

(1) If 〈p q〉 13 φ and 〈pprime qprime〉 extends 〈p q〉 then 〈pprime qprime〉 13 φ(2) No 〈p q〉 forces both φ and notφ(3) Every 〈p q〉 extends to 〈pprime qprime〉 such that 〈pprime qprime〉 13 φ or〈pprime qprime〉 13 notφ

(4) If 〈p q〉 13 φ and pprime p is bounded then 〈pprime q〉 13 φ

Proof (1) - (3) are immediate from the definition of forcing and (4)can be proved by induction on the complexity of φ

The following proposition establishes a relationship between thisforcing and ultrapowers

Proposition 46 (ldquo Losrsquos Theoremrdquo) Let φ(v1 vs) be an isin-formulawith parameters from VThen 〈p q〉 13 φ(Gn1

Gns) iff rank q = k gt n1 ns and

forallaai isin p forall〈x0 xkminus1〉 isin q(i) φ(xn1 xns

)

Proof For atomic formulas (cases (1) - (7)) the claim is immediatefrom the definition Case (9) is also trivial (union of two bounded setsis bounded)

Case (8) Let c = i isin p | forall〈x0 xkminus1〉 isin q(i) notφ(xn1 xns

)We need to prove that 〈p q〉 13 notφ iff p c is bounded

Assume that 〈p q〉 13 notφ and pc is unbounded We let pprime = pc andqprime(i) = 〈x0 xkminus1〉 isin q(i) | φ(xn1

xns) for i isin pprime qprime(i) = emptyk

for i isin I pprime Then 〈pprime qprime〉 isin H extends 〈p q〉 and by the inductiveassumption 〈pprime qprime〉 13 φ a contradiction

Conversely assume 〈p q〉 1 notφ and p c is bounded Then thereis 〈pprime qprime〉 of rank kprime extending 〈p q〉 such that 〈pprime qprime〉 13 φ By theinductive assumption there is a bounded set d such that

foralli isin (pprime d) forall〈x0 xkprimeminus1〉 isin qprime(i) φ(xn1

xns)


But (p c) cup d is a bounded set so there exist i isin (pprime cap c) d Forsuch i and 〈x0 xkprimeminus1〉 isin qprime(i) one has both notφ(xn1

xns) and

φ(xn1 xns

) a contradictionCase (10)

Let c = i isin p | forall〈x0 xkminus1〉 isin q(i) existv ψ(xn1 xns

v) We needto prove that 〈p q〉 13 existv ψ iff p c is bounded

Assume that 〈p q〉 13 existv ψ and p c is unbounded We let pprime = p cand qprime(i) = 〈x0 xkminus1〉 isin q(i) | not existv ψ(xn1

xns v) for i isin pprime

qprime(i) = emptyk for i isin I pprime Then 〈pprime qprime〉 extends 〈p q〉 and by thedefinition of 13 there exist 〈pprimeprime qprimeprime〉 extending 〈pprime qprime〉 with rank qprimeprime = kprimeprime

and m lt kprimeprime such that 〈pprimeprime qprimeprime〉 13 ψ(Gn1 Gns

Gm) By the inductiveassumption there is a bounded set d such that

foralli isin (pprimeprime d) forall〈x0 xkprimeprimeminus1〉 isin qprimeprime(i) ψ(xn1

xns xm)

Hence

foralli isin (pprimeprime d) forall〈x0 xkprimeprimeminus1〉 isin qprimeprime(i) existv ψ(xn1

xns v)

But i isin pprimeprime d implies i isin pprime this contradicts the definition of qprimeAssume that p c is bounded By the Reflection Principle in ZF

there is a least von Neumann rank Vα such that for all i isin c and all〈x0 xkminus1〉 isin q(i) there exists v isin Vα such that ψ(xn1

xns v)

Let 〈pprime qprime〉 be any condition extending 〈p q〉 and let kprime = rank qprime Welet pprimeprime = pprime cap c and

qprimeprime(i) =〈x0 xkprimeminus1 xkprime〉 |

〈x0 xkprimeminus1〉 isin qprime(i) and ψ(xn1

xns xkprime) and xkprime isin Vα

for i isin pprimeprime qprimeprime(i) = emptykprime otherwise Then 〈pprimeprime qprimeprime〉 extends 〈pprime qprime〉 and by

the inductive assumption 〈pprimeprime qprimeprime〉 13 ψ(Gn1 Gns

Gkprime) This provesthat 〈p q〉 13 existv ψ

We observe that if q is in Q and ℓ lt k = rank q then q ℓ definedby (q ℓ)(i) = 〈x0 xℓminus1〉 | existxℓ xkminus1 〈x0 xkminus1〉 isin q(i) isin Q

Corollary 47 If rank q = k gt n1 ns 〈pprime qprime〉 extends 〈p q〉 and

〈pprime qprime〉 13 φ(Gn1 Gns

) then 〈pprime qprime k〉 extends 〈p q〉 and 〈pprime qprime k〉 13φ(Gn1

Gns)

Lemma 48 Let z isin V For every 〈p q〉 there exist 〈p qprime〉 extending

〈p q〉 and m lt kprime = rank qprime such that 〈p qprime〉 13 z = Gm

Proof Let qprime(i) = 〈x0 xkminus1 xk〉 | 〈x0 xkminus1〉 isin q(i) and xk = zand m = k kprime = k + 1


We write 〈p q〉 13 z = Gm as 〈p q〉 13 z = Gm and 〈p q〉 13 z isin Gm

as 〈p q〉 13 z isin Gm We say that 〈p q〉 decides φ if 〈p q〉 13 φ or〈p q〉 13 notφ The following lemma is needed for the proof that theextended ultrapower satisfies SP

Lemma 49 For every 〈p q〉 and m lt k = rank q there is 〈pprime qprime〉 that

extends 〈p q〉 and is such that for every n isin N 〈pprime qprime〉 decides n isin Gm

Proof We first construct a sequence 〈〈pn qn〉 | n isin N〉 such that〈p0 q0〉 = 〈p q〉 and for each n rank qn = k pn+1 sub pn 〈pn+1 qn+1〉extends 〈pn qn〉 and 〈pn+1 qn+1〉 decides n isin Gm

Given pn let c = i isin pn | forall〈x0 xkminus1〉 isin qn(i) (n isin xm) Ifc is unbounded we let pprimen+1 = c and qn+1 = qn Otherwise pn c isunbounded and we let pprimen+1 = pn c and qn+1(i) = 〈x0 xkminus1〉 isinqn(i) | n isin xm for i isin pprimen+1 qn+1(i) = emptyk otherwise We obtainpn+1 from pprimen+1 by omitting the least element of pprimen+1 Proposition 46

implies that 〈pn+1 qn+1〉 decides n isin GmLet in be the least element of pn We define 〈pprime qprime〉 as follows i isin pprime

iff i isin pn and qprime(i) = qn(i) where in le i lt in+1 qprime(i) = emptyk otherwise

It is clear from the construction and Proposition 46 that 〈pprime qprime〉 isin H

it extends 〈p q〉 and it decides n isin Gm for every n isin N

42 Extended ultrapowers In this subsection we define the ex-tended ultrapower of a countable model of ZF by a generic filter U prove some fundamental properties of this structure and conclude thatit is a model of SPOT

We take Zermelo-Fraenkel set theory as our metatheory but theproof employs very little of its powerful machinery Subsection 85explains how the proof given below can be converted into a finitisticproof

We use ω for the set of natural numbers in the metatheory and r sas variables ranging over ω A set S is countable if there is a mappingof ω onto S

Let M = (MisinM) be a countable model of ZF Concepts definedin Subsection 41 make sense in M and all results of 41 hold in MWhen M is understood we use the notation and terminology from 41for the concepts in the sense of M thus N for NM ldquounboundedrdquo forldquounbounded in the sense of Mrdquo P for PM 13 for ldquo13 in the sense ofMrdquo etc The model M need not be well-founded externally and ω isisomorphic to an initial segment of N which may be proper

Definition 410 U subeM is a filter on P if

(1) M ldquop isin Prdquo for every p isin U


(2) If p isin U and M ldquopprime isin P and p extends pprimerdquo then pprime isin U (3) For eny p1 p2 isin U there is p isin U such that M ldquop extends p1 and

p extends p2rdquo

A filter U on P is M-generic if for every D isin M such that M

ldquoD is dense in Prdquo there is p isin U for which p isinM D

Since P has only countably many dense subsets in M M-genericfilters are easily constructed by recursion Let 〈pr | r isin ω〉 be anenumeration of P and 〈Dr | r isin ω〉 be an enumeration of all densesubsets of P in M Let q0 = p0 and for each s isin ω let qs+1 = prfor the least r such that M ldquopr extends qs and pr isin DsrdquoThen letU = p isinM | M ldquop isin P and qs extends prdquo for some s isin ωM-generic filters G sube M times M on H are defined and constructed

analogously

Lemma 411 If G is an M-generic filter on H then U = p isin P |existq 〈p q〉 isin G is an M-generic filter on P

Proof In M if D is dense in P then 〈p q〉 | p isin D and q isin Q is densein H

We now define the extended ultrapower of M by U we follow closelythe presentation in Spector [38]

Let Ω = m isin M | M ldquom isin Nrdquo We define binary relations =lowast

and isinlowast on Ω as followsm =lowast n iff there exists 〈p q〉 isin G such that rank q = k gt m n and

〈p q〉 13 Gm = Gnm isinlowast n iff there exists 〈p q〉 isin G such that rank q = k gt m n and〈p q〉 13 Gm isin Gn

It is easily seen from the definition of forcing and Proposition 46that =lowast is an equivalence relation on Ω and a congruence relation withrespect to isinlowast We denote the equivalence class of m isin Ω in the relation=lowast by Gm define Gm isin

lowast Gn iff m isinlowast n and let N = Gm | m isin ΩThe extended ultrapower of M by U is the structure N = (Nisinlowast)

There is a natural embedding j of M into N defined as follows ByLemma 48 for every z isin M there exist 〈p q〉 isin G and m lt rank q

such that 〈p q〉 13 z = Gm We let j(z) = Gm and often identify j(z)with z It is easy to see that the definition is independent of the choiceof representative from Gm and that j is an embedding of M into N

Proposition 412 (The Fundamental Theorem of Extended Ultrapow-ers) Let φ(v1 vs) be an isin-formula with parameters from M If Gn1

Gnsisin N then the following statements are equivalent

(1) N φ(Gn1 Gns

)


(2) There is some 〈p q〉 isin G such that 〈p q〉 13 φ(Gn1 Gns

)holds in M

(3) There exists some 〈p q〉 isin G with rank q = k gt n1 ns suchthat M foralli isin p forall〈x0 xkminus1〉 isin q(i) φ(xn1

xns)

Proof Statement (3) is just a reformulation of (2) using Proposition 46plus the fact that if d is bounded then 〈p q〉 isin G implies 〈(pd) q〉 isin G(Observe that D = 〈pprime qprime〉 | 〈pprime qprime〉 le 〈p q〉 and pprime le (p d) is densein 〈p q〉 for the definition of ldquodense inrdquo see the sentence precedingLemma 53)

The equivalence of (1) and (2) is the Forcing Theorem It is provedas usual by induction on the logical complexity of φ The cases v1 = v2and v1 isin v2 follow immediately from the definitions of =lowast and isinlowast andthe conjunction is immediate from (3) in the definition of a filter

We consider next the case where φ is of the form notψ First assumethat N notψ(Gn1

Gns) Lemma 45 (3) implies that there ex-

ists 〈p q〉 isin G such that either 〈p q〉 13 ψ(Gn1 Gns

) or 〈p q〉 13notψ(Gn1

Gns) In the first case N ψ(Gn1

Gns) by the induc-

tive assumption a contradiction Hence 〈p q〉 13 notψ(Gn1 Gns

)Assume that N 2 notψ(Gn1

Gns) then N ψ(Gn1

Gns)

and the inductive assumption yields 〈pprime qprime〉 isin G such that 〈pprime qprime〉 13ψ(Gn1

Gns) There can thus be no 〈p q〉 isin G such that 〈p q〉 13

notψ(Gn1 Gns

)Finally we assume that φ(u1 vs) is of the form existwψ(u1 vs w)

If N φ(Gn1 Gns

) then N ψ(Gn1 Gns

Gm) for some m isinΩ By the inductive assumption there is some 〈p q〉 isin G such that〈p q〉 13 ψ(Gn1

Gns Gm) and hence by the definition of forcing

〈p q〉 13 existwψ(Gn1 Gns

w)Conversely if 〈p q〉 isin G and 〈p q〉 13 existw ψ(Gn1

Gns w) then

by the definition of forcing there are 〈pprime qprime〉 isin G and m isin Ω such

that 〈pprime qprime〉 13 ψ(Gn1 Gns

Gm) By the inductive assumption N

ψ(Gn1 Gns

Gm) holds and hence N φ(Gn1 Gns

)

Corollary 413 The embedding j is an elementary embedding of M

into N

Corollary 414 The structure N satisfies ZF

Proposition 415 The structure N = (NisinlowastM) satisfies the princi-ples of Transfer Nontriviality and Standard Part

Proof Transfer is Corollary 413Working in M for every 〈p q〉 with rank q = k define qprime of rank k+1

by qprime(i) = 〈x0 xkminus1 xk〉 | 〈x0 xkminus1〉 isin q(i) and xk = i and


note that 〈p qprime〉 extends 〈p q〉 By M-genericity of G some such 〈p qprime〉belongs to G It is easily seen from the Fundamental Theorem that Gk

is an integer in N and that N ldquoGk 6= nrdquo for all n isin Ω Hence O

holds in NIt remains to prove the Standard Part principle Let Gm isin N By

Lemma 49 there is 〈p q〉 isin G which decides n isin Gm for all n isin Ω

The set E = n isin Ω | 〈p q〉 13 n isin Gm is definable in M hence thereis e isin M such that M ldquon isin erdquo iff n isin E iff N ldquon isin Gmrdquo ThusN ldquo st(e) and forallstν isin N (ν isin elarrrarr ν isin Gm)rdquo

This proves Proposition 42 and hence Theorem A

5 Conservativity of SCOT over ZFc

In this section we show that if ZF is replaced by ZFc the StandardPart principle can be strengthened to Countable st-isin-Choice

We recall that ZFc implies ACC this provides enough choice toprove that the ordinary ultrapower of a countable model M of ZFcby an M-generic filter U on P satisfies Losrsquos Theorem and thus yieldsan elementary extension of M A proof that every countable modelM = (MisinM) of ZFc has an extension to a model of SPOT + SC

in which M is the class of all standard sets can be obtained by astraightforward adaptation of the arguments in Enayatrsquos paper [8] inparticular of the proofs of Theorems B and C there Essentially allone has to do is to replace countable models of second-order arithmeticby countable models of ZFc We followed this approach in an earlyversion of the present paper

Another proof of conservativity of SCOT over ZFc was suggestedto us by Kanovei in a private communication Its basic idea is to useforcing to add to M a mapping of ω1 onto R without adding any realsIn the resulting generic extension there are nonprincipal ultrafilters overN and one can take an ultrapower of M by one of them to obtain anextension that satisfies SCOT However this method does not seemadequate for handling Idealization over uncountable sets in Section 7

We first outline the simplification of the forcing that is possible inthe presence of ACC and then use it to prove that assuming M is a

countable model of ZFc the structure N from Proposition 415 satisfiesalso CC and SN The proof can be viewed as a warm-up for similarbut more complex arguments of the following sections

We work in ZFc Given I = N and q isin Q of rank k ACC guar-antees the existence of a function f such that foralli isin I (f(i) isin q(i)) letq(i) = f(i) where f(i) = 〈f0(i) fkminus1(i)〉 For any 〈p q〉 isin H


the condition 〈p q 〉 extends 〈p q〉 We could replace Q by Q = q isinF | existk foralli isin I (q(i) isin Vk) But there is no need at all for the symbols

Gk k isin N and Spectorrsquos component Q of the forcing notion H if ourforcing language allows names for all f isin F (see below for details)

On the other hand Standardization and Countable st-isin-Choice un-like Transfer and Idealization deal with st-isin-formulas so we need toextend our definition of the forcing relation to such formulas

The forcing notion we use in this section is P The forcing language Lhas a constant symbol f for every function f isin F (the check is usuallysuppressed) Forcing is defined for arbitrary st-isin-formulas Only thefollowing clauses in the definiton of the forcing relation are necessary

Definition 51 (Simplified forcing)(1rsquo) p 13 f1 = f2 iff forallaai isin p (f1(i) = f2(i))(2rsquo) p 13 f1 isin f2 iff forallaai isin p (f1(i) isin f2(i))(8) p 13 notφ iff there is no pprime extending p such that pprime 13 φ(9) p 13 (φ and ψ) iff p 13 φ and p 13 ψ(10rsquo) p 13 existv ψ iff for every pprime extending p there exist pprimeprime extending pprime

and a function f isin F such that pprimeprime 13 ψ(f)(11)〈p q〉 13 st(f) iff existx forallaai isin p (f(i) = x) iff forallaai iprime isin p (f(i) = f(iprime))

The basic properties of forcing from Lemma 45 remain valid butProposition 46 (ldquo Losrsquos Theoremrdquo) of course holds only for isin-formulasin the form p 13 φ(f1 fr) iff forallaai isin p φ(f1(i) fr

(i))We now take Zermelo-Fraenkel set theory as our metatheory Let M

be a countable model of ZFc U an M-generic filter on P isin M andN = (Nisinlowast) the ultrapower of M by U If M ldquof isin Frdquo then [f ]Uis the equivalence class of f modulo U We identify x isin M with [cx]Uwhere cx is the constant function on I with value x in the sense of M

and let N = (NisinlowastM)Proposition 412 takes the following form

Proposition 52 Let φ(u1 ur) be an st-isin-formula with parame-ters from M If M ldquof1 fr isin Frdquo then the following statementsare equivalent

(1) N φ([f1]U [fr]U)(2) There is some p isin U such that p 13 φ(f1 fr) holds in M

Corollaries 413 and 414 and Proposition 415 remain valid in thismodified setting For isin-formulas Proposition 52 is just a fancy way tostate the ordinary Losrsquos Theorem but for st-isin-formulas it provides a

useful handle on the behavior of N


We need the following corollary (which can also be proved more te-diously directly from the definition of the forcing relation) The state-ment ldquoD sube P is dense in prdquo means that forallpprimeprime le p existpprime le pprimeprime (pprime isin D)

Lemma 53 If p 13 forallstm isin Nφ(m) then the set Dm = pprime isin P | pprime 13φ(m) is dense in p for every m isin N

Proof We show that the claim holds in every model M of ZFc Forevery pprimeprime le p in M there is an M-generic filter U such that pprimeprime isin U By

Proposition 52 N forallstmφ(m) so M ldquom isin Nrdquo implies N φ(m)By 52 again there exists pprime isin U such that pprime 13 φ(m) We can takepprime le pprimeprime

Lemma 54 Let φ(u v) be an st-isin-formula with parameters from LThen ZFc proves the following If p 13 forallstm existv φ(m v) then thereis pprime isin P and a sequence 〈fm | m isin N〉 such that pprime extends p andpprime 13 φ(m fm) for every m isin N

Proof By Lemma 53 the set Dm = pprime isin P | pprime 13 existv φ(m v) isdense in p for each m isin N Clause (10rsquo) in the definition of simplifiedforcing implies that also the set Em = pprime isin P | existf isin F pprime 13 φ(m f)is dense in p We let

〈mprime pprime〉R〈mprimeprime pprimeprime〉 iff pprimeprime sub pprime sube p and mprimeprime = mprime+1 and pprime isin Emprime and pprimeprime isin Emprimeprime

Applying ADC to the relation R we obtain a sequence 〈pm | m isin N〉such that p0 sube p and for each m pm+1 sub pm and existf isin F (pm 13

φ(m f)) We next use ACC to obtain a sequence 〈fm | m isin N〉 suchthat pm 13 φ(m fm) Note that the Reflection Principle of ZF providesa set A = Vα such that for all m

(existf isin F pm 13 φ(m f))rarr (existf isin F capA pm 13 φ(m f))

As in the proof of Lemma 49 let im be the least element of pm andlet pprime =

⋃infin

m=0 pm cap (im+1 im) Then for every m the set pprime pm isbounded hence pprime 13 φ(m fm)

Proposition 55 If M satisfies ZFc then N satisfies CC

Proof Assume that N forallstm existv φ(m v) Then there is p isin U suchthat p 13 forallstm existv φ(m v) By Lemma 54 there is pprime isin U and a sequence〈fm | m isin N〉 such that pprime 13 φ(m fm) for every m isin N

We define a function g on I by g(i) = 〈m fm(i)〉 | m isin NRecall that g is the name for g in the forcing language By LosrsquosTheorem pprime 13 ldquog is a function with domain Nrdquo and for every m isin N

pprime 13 g(m) = fm We conclude that N ldquog is a function with domain

Nrdquo and N forallstm isin Nφ(m g(m))


Proposition 56 If M satisfies ZFc then N satisfies CCst

Proof The assumption N forallstm existstv φ(m v) implies that we can takefm = cxm

in Lemma 54 and the proof of Proposition 55 For thefunction g on I defined by g(i) = 〈m xm〉 | m isin N we then have

N ldquog is standardrdquo

Let p pprime isin P and let γ be an increasing mapping of pprime onto p weextend γ to I = N by defining γ(a) = 0 for a isin I p

Lemma 57 p 13 φ(f1 fr) iff pprime 13 φ(f1 γ fr γ)

Proof This follows by induction on the cases in the definition of simpli-fied forcing using the observation that the mapping pprimeprime rarr pprime cap γminus1[pprimeprime]is an isomorphism of the posets (pprimeprime isin P | pprimeprime le ple) and (pprimeprime isin P |pprimeprime le pprimele)

Corollary 58 Let φ be an st-isin-formula with parameters from VThen p 13 φ iff pprime 13 φ

Proposition 59 The structure N satisfies the principle of Standard-ization for st-isin-formulas with no parameters

Proof For standard x N φ(x) iff p 13 φ(x) for every p isin P Theright side is expressible by an isin-formula

This completes the proof of Theorem D

Another principle that can be added to SCOT is Dependent Choicefor st-isin-formulas

DC Let φ(u v) be an st-isin-formula with arbitrary parametersIf B is a set b isin B and forallx isin B existy isin B φ(x y) then there is a sequence〈bn | n isin N〉 such that b0 = b and forallstn isin N (bn isin B and φ(bn bn+1))

Theorem 510 SCOT + DC is a conservative extension of ZFc

Proof We show that DC holds in the structure NLet M ldquob B isin Frdquo and p 13 b isin B and forallx isin B existy isin B φ(x y) We

now let A = 〈pprime f prime〉 | pprime le p and pprime 13 f prime isin B note that 〈p b〉 isin Aand define R on A by 〈pprime f prime〉R〈pprimeprime f primeprime〉 iff pprimeprime le pprime and pprimeprime 13 φ(f prime f primeprime)

It is clear from the properties of forcing that for every 〈pprime f prime〉 isin Athere is 〈pprimeprime f primeprime〉 isin A such that 〈pprime f prime〉R〈pprimeprime f primeprime〉 Using ADC we obtaina sequence 〈〈pn fn〉 | n isin N〉 such that p0 le p f0 = b and for all n isin N〈pn fn〉 isin A pn+1 le pn and pn+1 13 φ(fn fn+1)

The rest of the proof imitates the arguments in the last paragraphsof the proofs of Lemma 54 and Proposition 55


6 Standardization for parameter-free formulas

In this section we prove that the structure N = (NisinlowastM) satisfiesthe principle of Standardization for parameter-free formulas assumingonly that the model M satisfies ZF Explicitly the principle postulates

SN Let φ(v) be an st-isin-formula with no parameters Then

forallstA existstS forallstx (x isin S larrrarr x isin A and φ(x))

Lemma 61 The principle SN is equivalent to Standardization forst-isin-formulas with standard parameters

Proof Given φ(v p1 pℓ) where p1 pℓ are standard we let P =〈p1 pℓ〉 and apply SN to the formula ψ(w) with no parametersexpressing ldquoexistw1 wℓ (w = 〈v w1 wℓ〉 and φ(v w1 wℓ))rdquo and tothe standard set AtimesP We get a standard S such that for all standardinputs 〈x y1 yℓ〉 isin S larrrarr 〈x y1 yℓ〉 isin AtimesP and φ(x y1 yℓ)holds The set T = x isin A | 〈x p1 pℓ〉 isin S standardizesφ(v p1 pℓ)

With the exception of the last proposition in this section we work inZF As in Section 5 we begin with extending forcing to st-isin-formulas

We add the following clauses to Definition 44(11) 〈p q〉 13 st(z) for every z isin V

(12) 〈p q〉 13 st(Gn) iff rank q = k gt n and

existx forallaai isin p forall〈x0 xkminus1〉 isin q(i) (xn = x) or equivalently

forallaai iprime isin p forall〈x0 xkminus1〉 isin q(i) forall〈xprime0 x

primekminus1〉 isin q(i

prime) (xn = xprimen)

The basic properties of forcing from Lemma 45 in Subsection 41remain valid but ldquo Losrsquos Theoremrdquo does not hold for st-isin-formulasHowever Proposition 46 is instrumental in the proof of Lemma 49The technical lemma that follows is a simple consequence of Proposi-tion 46 for forcing of isin-formulas but it remains valid even for forcingof st-isin-formulas

Definition 62 For σ = 〈n1 ns〉 where n1 ns lt k are mutu-ally distinct let πk

σ Vk rarr Vs be the ldquoprojectionrdquo of Vk onto Vs

πkσ(〈x0 xkminus1〉) = 〈xn1

xns〉

For q isin Q of rank k q σ of rank s is defined by (q σ)(i) = πkσ[q(i)]

Lemma 63 Let φ be an st-isin-formula with parameters from VAssume that rank q1 = k1 σ1 = 〈n1 ns〉 where n1 ns lt k1and rank q2 = k2 σ2 = 〈m1 ms〉 with m1 ms lt k2 If


(q1 σ1)(i) = (q2 σ2)(i) for all i isin p (we write q1 σ1 =p q2 σ2) then

〈p q1〉 13 φ(Gn1 Gns

) if and only if 〈p q2〉 13 φ(Gm1 Gms

)

Proof The proof is by induction on the logical complexity of φFor isin-formulas the assertion follows immediately from Proposition 46

In particular it holds for all atomic formulas involving isin and = (cases(1) - (7) in the definition of forcing) It is also clear for st (cases (11)and (12)) and for conjunction (case (9))

Case (8)

Assume that the statement is true for φ 〈p q1〉 13 notφ(Gn1 Gns

)

and 〈p q2〉 1 notφ(Gm1 Gms

) Then there exists a condition 〈pprime qprime2〉 le〈p q2〉 such that 〈pprime qprime2〉 13 φ(Gm1

Gms) From the inductive as-

sumption it follows that 〈pprime qprime2 σ2〉 13 φ(G0 Gsminus1) (recall thatqprime2 σ2 sube Vs) Let q = qprime2 σ2 le q2 σ2 =p q1 σ1 We defineqprime1 = (πk1

σ1)minus1[q] cap q1 le q1 Now 〈pprime qprime1〉 le 〈p q1〉 and qprime2 σ2 =pprime q

prime1 σ1

so by the inductive assumption 〈pprime qprime1〉 13 φ(Gn1 Gns

) This is acontradiction with 〈p q1〉 13 notφ(Gn1

Gns) The reverse implication

follows by exchanging the roles of 〈p q1〉 and 〈p q2〉Case (10)Assume that 〈p q1〉 13 existv ψ(Gn1

Gns v) Let 〈pprime qprime2〉 le 〈p q2〉

where rank qprime2 = kprime2 = k2 + r We need to find 〈pprimeprime qprimeprime2〉 extending 〈pprime qprime2〉and m such that 〈pprimeprime qprimeprime2〉 13 ψ(Gm1

Gms Gm) This will prove that

〈p q2〉 13 existv ψ(Gm1 Gms

v)

We let q = πkprime2σ2 [qprime2] sube πk2

σ2[q2] =p π

k1σ1

[q1] and macrq = (πk1σ1

)minus1[q] cap q1 le q1We define qprime1 of rank kprime1 = k1 + r byqprime1(i) = 〈x0 xk1minus1 y0 yrminus1〉 | 〈x0 xk1minus1〉 isin macrq and

〈xn1 xns

〉 = πkprime2

σ2(〈xprime0 xprimek2minus1 y0 yrminus1〉) for some

〈xprime0 xprimek2minus1 y0 yrminus1〉 isin q

prime2

for i isin pprime qprime1(i) = emptykprime1 otherwise We observe that qprime1(i) 6= empty and

πkprime1σ1 [q

prime1] =pprime π

kprime2σ2 [qprime2]

We have 〈pprime qprime1〉 le 〈p q1〉 hence there are 〈pprimeprime qprimeprime1〉 le 〈pprime qprime1〉 with

rank qprimeprime1 = kprimeprime1 and n such that 〈pprimeprime qprimeprime1〉 13 ψ(Gn1 Gns

Gn) Finallywe construct qprimeprime2 of rank kprimeprime2 = kprime2 + 1 such that 〈pprimeprime qprimeprime2〉 le 〈p

primeprime qprime2〉 and

for some m πkprimeprime1σ1n(qprimeprime1) =pprimeprime π

kprimeprime2σ2m(qprimeprime2) where σ1n = 〈n1 ns n〉

and σ2m = 〈m1 ms m〉 By the inductive assumption this es-

tablishes 〈pprimeprime qprimeprime2〉 13 ψ(Gm1 Gms

Gm)

We start with q = πkprimeprime1

σ1 [qprimeprime1 ] sube πkprime1

σ1 [qprime1] =pprime πkprime2

σ2 [qprime2] and define

qprimeprime2(i) = 〈x0 xkprime2minus1 z〉 | 〈x0 xkprime

2minus1〉 isin (π

kprime2

σ2)minus1[q ] cap qprime2 and

〈xn1 xns

z〉 isin πkprimeprime1

σ1n(qprimeprime1)


for i isin pprimeprime qprimeprime2(i) = emptykprime2+1 otherwise

We have 〈pprimeprime qprimeprime2〉 le 〈pprimeprime qprime2〉 and rank qprimeprime2 = kprimeprime2 = kprime2 + 1 Let m = kprime2 It

follows from the construction that πkprimeprime1σ1n(qprimeprime1 ) =pprimeprime π

kprimeprime2σ2m(qprimeprime2)

Corollary 64 Let φ be an st-isin-sentence with parameters from VThen 〈p q〉 13 φ iff 〈p 1〉 13 φ

As in Section 5 let p pprime isin P and let γ be an increasing mapping ofpprime onto p we extend γ to I by defining γ(a) = 0 for a isin I p

Lemma 65 Let φ(v1 vs) be an st-isin-formula with parametersfrom V Then 〈p q〉 13 φ(Gn1

Gns) iff 〈pprime q γ〉 13 φ(Gn1

Gns)

Proof As for Lemma 57

Corollary 66 Let φ(v1 vr) be an st-isin-formula For z1 zr isin V〈p q〉 13 φ(z1 zr) iff 〈pprime qprime〉 13 φ(z1 zr)

Proposition 67 The structure N = (NisinlowastM) satisfies the principleof Standardization for st-isin-formulas with no parameters

Proof For standard z N φ(z) iff 〈p q〉 13 φ(z) for every 〈p q〉 isin HThe right side is expressible by an isin-formula

This completes the proof of Theorem B

There is yet another principle that can be added to SPOT and keepit conservative over ZF One of its important consequences is the im-possibility to uniquely specify an infinitesimal

UP (Uniqueness Principle) Let φ(v) be an st-isin-formula with stan-dard parameters If there exists a unique x such that φ(x) then this xis standard

Theorem 68 SPOT + UP is a conservative extension of ZF

Proof If N existx [φ(x) and forally (φ(y) rarr y = x) and not st(x)] then there is

〈p q〉 isin G and m lt k = rank q such that 〈p q〉 13 φ(Gm) and not st(Gm) andforally (φ(y)rarr y = Gm)

Let r = q m ie for all i isin I x isin r(i) larrrarr exist〈x0 xkminus1〉 isinq(i) (xm = x)

Claim 1 For every x isin⋃

iisinp r(i) the set px = i isin p | x isin r(i) isbounded we let ix denote its greatest element

Proof of Claim 1 Let x be such that px is unbounded Define

qx(i) = 〈x0 xkminus1〉 isin q(i) | (xk = x) for i isin px

= emptyk otherwise


Then 〈px qx〉 le 〈p q〉 and 〈px qx〉 13 Gm = x ie 〈px qx〉 13 st(Gm) acontradiction

Claim 2 There exist unbounded mutually disjoint sets p1 p2 sub p andnonempty sets s(i) sube r(i) for all i isin p1 cup p2 such that(⋃

iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty

We postpone the proof of Claim 2 and complete the proof of thetheorem

We let q(i) = 〈x0 xkminus1〉 isin q(i) | xk isin s(i) for i isin p1 cup p2q(i) = emptyk otherwise We have 〈p1 q〉 le 〈p q〉〈p2 q〉 le 〈p q〉 andconsequently 〈p1 q〉 13 φ(Gm) 〈p2 q〉 13 φ(Gm)

Let γ be an increasing mapping of p1 onto p2 extended by γ(i) = 0for i isin I p1 By Lemma 65 〈p1 q γ〉 13 φ(Gm) We ldquoamalga-materdquo 〈p1 q〉 and 〈p1 q γ〉 to form a condition of rank 2k as follows〈x0 xkminus1 y0 ykminus1〉 isin q(i)larrrarr

〈x0 xkminus1〉 isin q(i) and 〈y0 ykminus1〉 isin q(γ(i))

and observe that 〈p1 q〉 le 〈p q〉 Let σ1 = k and σ2 = k+ ℓ | ℓ lt kWe have q σ1 = q and q σ2 = q γ By Lemma 63 〈p1 q〉 13 φ(Gm) andφ(Gk+m) But xm 6= ym holds for all 〈x0 xkminus1 y0 ykminus1〉 isin q(i)and all i isin p1 so 〈p1 q〉 13 Gm 6= Gk+m This contradicts 〈p1 q〉 13forally (φ(y)rarr y = Gm)

Proof of Claim 2 Wlog we can assume p = I = N (map p ontoN in an increasing way) Define sequences 〈nℓ | ℓ isin N〉〈αℓ | ℓ isin N〉and 〈s(nℓ) | ℓ isin N〉 by recursion as follows

Let n0 = 0 α0 = minix | x isin r(0) and s(n0) = s(0) = x isin r(0) |ix = α0

At stage ℓ + 1 let nℓ+1 = αℓ + 1 αℓ+1 = minix | x isin r(nℓ+1) ands(nℓ+1) = x isin r(nℓ+1) | ix = αℓ+1

We observe that s(nℓ) cap s(nℓprime) = empty for all ℓ 6= ℓprime It remains to letp1 = n2ℓ | ℓ isin N and p2 = n2ℓ+1 | ℓ isin N

7 Idealization

We recall the axioms of the theory BST see the references Kanoveiand Reeken [24] and Fletcher et al [9] for motivation and more detailIn addition to the axioms of ZFC they are

B (Boundedness) forallx existsty (x isin y)

T (Transfer) Let φ(v) be an isin-formula with standard parametersThen



S (Standardization) Let φ(v) be an st-isin-formula with arbitrary pa-rameters Then


BI (Bounded Idealization) Let φ be an isin-formula with arbitraryparameters For every set A

forallst fina sube A existy forallx isin a φ(x y)larrrarr existy forallstx isin A φ(x y)

71 Idealization over uncountable sets In order to obtain modelswith Bounded Idealization the construction of Subsections 41 and 42needs to be generalized from I = N to I = Pfin(A) where A is anyinfinite set The key is the right definition of ldquounboundedrdquo subsets ofI We work in ZF

We use the notation Plem(A) for a isin Pfin(A) | |a| le m

Definition 71 A set p sube Pfin(A) is thick if

forallm isin N existn isin N foralla isin Plem(A) existb isin p cap Plen(A) (a sube b)

We let νp(m) denote the least n with this property The set p is thinif it is not thick

Clearly a isin Pfin(A) | x isin a is thick for every x isin A (with νp(m) =m+ 1) We now carry out the developments of Subsections 41 and 42with unbounded and bounded replaced by thick and thin respectivelyThe definition of forcing and proofs of Lemmas 45 and 48 are asbefore The following observation enables the proof of Proposition 46to go through as well

Lemma 72 If p is thick and S sube Pfin(A) then either p cap S or p Sis thick

Proof Otherwise there is m1 such that

() foralln isin N exista1 isin Plem1(A) forallb isin (p cap S) cap Plen(A) (a1 b)

and there is m2 such that

() foralln isin N exista2 isin Plem2(A) forallb isin (p S) cap Plen(A) (a2 b)

Let m1 m2 be as above and let m = m1 +m2 Since p is thick

() existn isin N foralla isin Plem(A) existb isin p cap Plen(A) (a sube b)

Fix such n for a1 isin Plem1(A) such that forallb isin (pcapS)capPlen(A) (a1 b)

and a2 isin Plem2(A) such that forallb isin (p S) cap Plen(A) (a2 b) we have

a1cupa2 isin Plem(A) By () there is b isin pcapPlen(A) such that a1cupa2 sube b

Depending on whether b isin p cap S or b isin p S this contradicts () or()


The next lemma enables a generalization of Lemma 49

Lemma 73 Let 〈pn | n isin N〉 be such that for all n pn isin P andpn supe pn+1 Then there is p isin P with the property that for every n thereis k isin N such that foralla isin (p pn) (|a| le k) In particular p pn is thin

Proof Define p =⋃infin

m=0a isin pm | |a| le νpm(m) Since p pn sube⋃nminus1m=0a isin pm | |a| le νpm(m) a isin p pn implies |a| le k for k =

maxνp0(0) νpnminus1(nminus 1)

We show that p is thick Given m isin N we let n = νpm(m) Ifa isin Pfin(A) and |a| le m then there is b isin pm such that a sube band |b| le n By the definition of p b isin p So n has the requiredproperty

With these changes the rest of the development of Subsections 41and 42 goes through and establishes the following strengthening ofProposition 415

Proposition 74 Assume that M ldquoI = Pfin(A) and A is infiniterdquo

The structure NA = (NAisinlowastM) constructed for this I satisfies the

principles of Transfer Nontriviality Boundedness Standard Part andBounded Idealization over A for isin-formulas with standard parameters

Proof To prove that NA O one can take d isinM such that M ldquod isa function on I = Pfin(A) and foralla isin Pfin(A) (d(a) = |a|)rdquo Nontrivialityalso follows from Bounded Idealization

Let Gm isin N and let 〈p q〉 isin G have rank q = k gt m There is someX isin M such that M ldquoforalli isin I forall〈x0 xkminus1〉 isin q(i) (xm isin X)rdquo ByProposition 46 〈p q〉 13 forallv (v isin Gm rarr v isin X) so forallv isin Gm (v isin X)

holds in NA This proves Boundedness in NA

It remains to prove that Bounded Idealization over A holds in NALet φ(u v) be an isin-formula with parameters from M Assume thatM foralla isin Pfin(A) existy forallx isin a φ(x y) By the Reflection Principle inZF M ldquoexist an ordinal α foralla isin Pfin(A) existy isin Vα forallx isin a φ(x y)rdquo Wework in the model M

Let 〈p q〉 isin H be a forcing condition and k = rank q For i = a isin pwe let qprime(a) =

〈x0 xkminus1 xk〉 | 〈x0 xkminus1〉 isin q(a)and xk isin Vα andforallx isin a φ(x xk)

qprime(a) = emptyk+1 otherwise Then 〈p qprime〉 extends 〈p q〉 For every x isin Athe set c = p cap a isin Pfin(A) | x isin a is in P because p c is thin

(Lemma 72) and so by Proposition 46 〈p qprime〉 13 φ(x Gk)By the genericity of G there exist 〈p q〉 isin G and k isin Ω such that

〈p q〉 13 φ(x Gk) for all x isin A By the Fundamental Theorem 412


NA φ(xGk) for all x isin A This means that the rarr implication ofIdealization over A for isin-formulas with standard parameters is satisfied

in NA The opposite implication follows from SP see Lemma 21 inSection 2

72 Further theories Nelsonrsquos IST postulates a form of Idealizationthat is even stronger than Bounded Idealization (but it contradictsBoundedness)I (Idealization) Let φ be an isin-formula with arbitrary parameters

forallst fina existy forallx isin a φ(x y)larrrarr existy forallstx φ(x y)

The theory BSPTprime = ZF + T + SPprime + B + BIprime is introduced inSubsection 13 We let ISPTprime be the theory obtained from BSPTprime bydeleting Boundedness and replacing Bounded Idealization for formulaswith standard parameters by Iprime Nelsonrsquos Idealization for formulas withstandard parameters In other words ISPTprime = ZF + T + SPprime + Iprime

Principle I implies the existence of a finite set that contains all stan-dard sets as elements and has certain undesirable consequences fromthe metamathematical point of view Kanovei and Reeken [24] The-orem 4623 prove that there are countable models M = (MisinM) ofZFC that cannot be extended to a model of IST in which M would bethe class of all standard sets (assuming ZFC is consistent) We do notknow whether the same is the case for ISPTprime Nevertheless we havethe following result

Theorem 75 ISPTprime is a conservative extension of ZF

Proof Let us assume that ISPTprime ⊢ θ but ZF 0 θ for some isin-sentenceθ Let ZFr be ZF with the Axiom Schema of Replacement restrictedto Σr-formulas and let ISPTprime

r be ISPTprime with ZF replaced by ZFrThere is r isin ω for which ISPTprime

r ⊢ θLet M = (MisinM) be a model of ZF+notθ By the Reflection Principle

of ZF valid in M there is α isin M such that M ldquoα is a limit ordinalrdquoM φVα for every axiom φ of ZFr and M (notθ)Vα

We let A = (Vα)M and use Proposition 74 to extend M to a model

NA We define Nα = x isin NA | NA x isin Vα It is easy to verify

that Nα = (NαisinlowastNαM cap Nα) is a model of ISPTprime

r hence θ holds

in Nα On the other hand (notθ)Vα holds in M and hence by Transfer

notθ holds in Nα A contradiction

Kanovei and Reeken [24] Theorem 345 showed that the class ofbounded sets in IST satisfies the axioms of BST This result holdsalso for ISPTprime and BSPTprime respectively and establishes the followingtheorem


Theorem 76 The theory BSPTprime is a conservative extension of ZF

This concludes the proof of Theorem C

Finally we prove that if M satisfies ADC then the model NA con-structed in Proposition 74 satisfies CC We note that the definitionof forcing for st-isin-formulas in Section 6 and Lemma 63 extend toI = Pfin(A)

Proposition 77 If M is a countable model of ZFc then the extended

ultrapower NA satisfies Countable st-isin-Choice (both CC and CCst)

Proof We work in ZFcCC Let us assume that 〈p q〉 has rank k and 〈p q〉 13 forallstm isin

N existv φ(m v) Let Em =

〈pprime qprime〉 isin H | 〈pprime qprime〉 le 〈p q〉 and 〈pprime qprime〉 13 φ(m Gn) for some n gt k

By an argument like the one in Lemma 53 it follows that for every mand every 〈pprimeprime qprimeprime〉 le 〈p q〉 there is 〈pprime qprime〉 le 〈pprimeprime qprimeprime〉 such that 〈pprime qprime〉 isinEm [We note that the Em may be proper classes but by the ReflectionPrinciple there is a set S such that 〈p q〉 isin S and for every m and every〈pprimeprime qprimeprime〉 isin S there is 〈pprime qprime〉 le 〈pprimeprime qprimeprime〉 such that 〈pprime qprime〉 isin S cap Em Theclasses Em can be replaced by the sets S capEm in the argument below]

We let 〈mprime 〈pprime qprime〉〉R〈mprimeprime 〈pprimeprime qprimeprime〉〉 iff

〈pprimeprime qprimeprime〉 le 〈pprime qprime〉 le 〈p q〉 andmprimeprime = mprime+1and 〈pprime qprime〉 isin Emprime and 〈pprimeprime qprimeprime〉 isin Emprimeprime

Applying ADC to the relation R we obtain a sequence 〈〈pm qm〉 |m isin N〉 such that 〈p0 q0〉 le 〈p q〉 and for each m 〈pm+1 qm+1〉 le〈pm qm〉 and 〈pm qm〉 13 φ(m Gn) for some n isin N n gt k Letrank qm = ℓm and let nm lt ℓm nm gt k be the least such n

As in the proof of Lemma 73 let pinfin =⋃infin

m=0Cm where Cm = a isinpm | |a| le νpm(m) We recall that pinfin pm is thin for every m hence

〈pinfin qm〉 13 φ(m Gnm) We define a function qinfin isin Q of rank k + 1 as

follows If a isin Cm ⋃mminus1

j=0 Cj then

qinfin(a) = 〈x0 xk〉 | xk is a function and dom xk = N and forallj le m

exist yk yℓjminus1 (〈x0 xkminus1 yk yℓjminus1〉 isin qj(a) and xk(j) = ynj)and

forallj gt m (xk(j) = 0)

By ldquo Losrsquos Theoremrdquo 〈pinfin qinfin〉 13 ldquoGk is a function with dom Gk =Nrdquo

Now assume that NA forallstm existv φ(m v) Then there is 〈p q〉 isin Gsuch that 〈p q〉 13 forallstm existv φ(m v) By the above discussion thereis a condition of the form 〈pinfin qinfin〉 such that 〈pinfin qinfin〉 isin G hence


NA ldquoGk is a function with domGk = Nrdquo Fix m isin M let NA

Gk(m) = Gℓ we can assume ℓ gt k Then there is some 〈pprime qprime〉 isin G

〈pprime qprime〉 le 〈pinfin qinfin〉 such that 〈pprime qprime〉 13 Gk(m) = GℓLet σm = k cup nm and σprime = k cup ℓFrom 〈pinfin qm〉 13 φ(m Gnm

) and Lemma 63 it follows that the con-dition 〈pinfin qm σm〉 13 φ(m Gk) We see from the construction that〈pprime cap pm q

prime σprime〉 le 〈pprime cap pm qm σm〉 As pprime pm is thin we have

〈pprime qprime σprime〉 13 φ(m Gk) and by Lemma 63 again 〈pprime qprime〉 13 φ(m Gℓ)

From 〈pprime qprime〉 isin G we conclude that NA φ(mGℓ) and hence also

NA φ(mGk(m))CCst Assuming 〈p q〉 13 forallstm isin N existstv φ(m v) we can require in

the definition of Em that 〈pprime qprime〉 13 st(Gn) ie 〈pprime qprime〉 13 Gn = cznfor a uniquely determined zn In the definition of qinfin(a) we can let

xk = 〈znj| j isin N〉 Then 〈pinfin qinfin〉 13 st(Gk) and NA st(Gk)

Let BSCTprime be the theory obtained from BSPTprime by adding ADC

and strengthening SP to CC analogously for ISCTprime The last twotheorems of this section follow by the same arguments as those used toprove Theorems 75 and 76

Theorem 78 The theory ISCTprime is a conservative extension of ZFc

Theorem 79 The theory BSCTprime is a conservative extension of ZFc

8 Final Remarks

81 Open problems

(1) Are the theories BSCTprime +SN and ISCTprime +SN (defined above)conservative extensions of ZFc

We do not know whether NA for I = Pfin(A) with uncountable Asatisfies SN In the absence of AC a way to formulate and prove asuitable analog of Lemma 57 is not obvious

(2) Are the theories BSPT and ISPT conservative extensions ofZF Are the theories BSCT and ISCT conservative extensions ofZFc

Here BSPT is obtained from BSPTprime by strengthening (Bounded)Idealization to allow arbitrary parameters similarly for the other the-ories The likely answer is yes the obvious approach is to iterate theforcing used to prove the primed versions Spector develops iterated ex-tended ultrapowers in [39] His method would require nontrivial adap-tations in our framework but it is likely to work provided the answerto problem (1) is yes The ultimate result would be that BSCT+SN

and ISCT + SN are conservative extensions of ZFc


(3) Does every countable model of ZF have an extension to a modelof BSPTprime

The likely answer is again yes using a suitable iteration of extendedultrapowers

(4) Is SPOT + SC a conservative extension of ZF

82 Forcing with filters A more elegant and potentially more pow-erful notion of forcing is obtained by replacing P with

P = P | P is a filter of unbounded subsets of I

where P prime extends P iff P sube P prime ldquo Losrsquos Theoremrdquo 46 then takes theform 〈P q〉 13 φ(Gn1


existp isin P foralli isin p forall〈x0 xkminus1〉 isin q(i) φ(xn1 xns

)The forcing notion P we actually use amounts to restricting oneself toprincipal filters

83 Zermelo set theory Similar results can be obtained for theo-ries weaker than ZF Let Z = ZF minus Replacement be the Zermeloset theory and let BT denote Transfer for bounded formulas In theproof of Proposition 42 the extended ultrapower can be replaced bythe extended bounded ultrapower (see Chang and Keisler [6] Sec 44for a discussion of ordinary bounded ultrapowers) This proves thatSPOTminus = Z + O + BT + SP is a conservative extension of Z Withsome modifications this theory can be taken as an axiomatization ofthe internal part of nonstandard universes of Keisler [6 27] (the super-structure framework for nonstandard analysis) Analogous results canbe obtained for SCOTminus BSPTminus and BSCTminus

84 Weaker theories Reverse Mathematics has as its goal the cal-ibration of the exact set-theoretic strength of the principal results inordinary mathematics One of its chief accomplishments is the discov-ery that with a few exceptions every theorem in ordinary mathematicsis logically equivalent (over S1) to one of the five subtheories S1 minus S5

of second-order arithmetic Z2 (Simpson [35 p 33]) known collectivelyas ldquoThe Big Fiverdquo Here S1 is the weakest of the five theories thesecond-order arithmetic with recursive comprehension axiom also de-noted RCA0 and S2 known as WKL0 is obtained by adding theWeak Konigrsquos Lemma to the axioms of RCA0 We refer to Simp-son [35] for a comprehensive introduction to Reverse Mathematics

Keisler and others extended the ideas of Reverse Mathematics tothe nonstandard realm In Keislerrsquos paper [28] it is established that ifS is any of the ldquoBig Fiverdquo theories above then S has a conservativeextension lowastS to a theory in the language with an additional unary


predicate st the axioms of lowastS include O SP and for the theoriesstronger than WKL0 also FOT (First-Order Transfer)

In a somewhat different direction there is an extensive body of workby van den Berg Sanders and others (see [5] [32] and the referencestherein) devoted to determining the exact proof-theoretic strength ofparticular results in infinitesimal ordinary mathematics Substantialparts of it can be carried out in these and other elementary systemsfor nonstandard mathematics for example Nelson [30] and Sommerand Suppes [37] However these systems do not enable the naturalreasoning as practiced in analysis They are usually formalized in thelanguage of second-order arithmetic or type theory Basic objects ofordinary analysis such as real numbers continuous functions and sepa-rable metric spaces have to be represented in these theories via suitablecodes and the results may have to be presented ldquoup to infinitesimalsrdquobecause the full strength of the Transfer principle or the Standard Partprinciple is not available The focus of this paper is on theories likeSPOT or SPOTminus which axiomatize both the traditional and thenonstandard methods of ordinary mathematics in the way they arecustomarily practiced Rather than looking for the weakest principlesthat enable a proof of a given mathematical theorem we formulatetheories that are as strong as possible while still effective (conservativeover ZF) or semi-effective (conservative over ZFc)

85 Finitistic proofs The model-theoretic proof of Proposition 42as given here is carried out in ZF Using techniques from Simpson [35]Chapter II esp II3 and II8 it can be verified that the proof goesthrough in RCA0 (wlog one can assume that M sube ω)

The proof of Theorem A from Proposition 42 requires the GodelCompleteness Theorem and therefore WKL0 see [35] Theorem IV33We conclude that Theorem A can be proved in WKL0

Theorem A when viewed as an arithmetical statement resulting fromidentifying formulas with their Godel numbers is Π0

2 It is well-knownthat WKL0 is conservative over PRA (Primitive Recursive Arith-metic) for Π0

2 sentences ([35] Theorem IX316) therefore Theorem A

is provable in PRA The theory PRA is generally considered to cor-rectly capture finitistic reasoning as envisioned by Hilbert [12] (seeeg Simpson [35] Remark IX318) and HilbertndashBernays [13 14] (seeZach [42] p 417) We conclude that Theorem A has a finitistic proof

These remarks apply equally to Theorems B - D

86 SPOT and CH Connes (see for example [7] pp 20ndash21) objectsto the use of ultrafilters but approves of the Continuum Hypothesis


(CH) In the absence of full AC it is important to distinguish (atleast) two versions of CHCH Every infinite subset of R is either countable or equipotent

to RCH+ R is equipotent to alefsym1 (often written 2alefsym0 = alefsym1)The axioms CH and CH+ are equivalent over ZFC but not over

ZFc It is known that ZFc+ CH does not imply the existence of anynonprincipal ultrafilters over N (CH holds in the Solovay model) Wehave

Proposition 81 The theory SPOT+CH is a conservative extensionof ZF + CH

Proof Let φ be an isin-sentence Then SPOT + CH ⊢ φ iff SPOT ⊢(CHrarr φ) iff ZF ⊢ (CHrarr φ) iff ZF + CH ⊢ φ

However it seems clear that Connes has CH+ in mind But CH+

implies that R has a well-ordering (of order type alefsym1) From this iteasily follows that there exist nonprincipal ultrafilters over N (for ex-ample Jech [21] p 478 proves a much stronger result) Thus Connesrsquosposition on this matter is incoherent

Apart from the issue of CH Connesrsquos repeated criticisms of Robin-sonrsquos framework starting in 1994 are predicated on the premise thatinfinitesimal analysis requires ultrafilters on N (which are incidentallyfreely used in some of the same works where Connes criticizes Robin-son) Our present article shows that Connesrsquos premise is erroneousfrom the start4

87 External sets This paper employs only definable external setsand only in Subsection 3 It is sometimes claimed that the axiomaticapproach is inferior to the model-theoretic one because substantial useof external sets is essential for some of the most important new con-tributions of Robinsonian nonstandard analysis to mathematics suchas the constructions of nonstandard hulls and Loeb measures Thefollowing observations are relevant

bull Except in some very special cases nonstandard hulls and Loebmeasures fall in the scope of set-theoretic mathematics and theuse of AC in their construction is not an issuebull Hrbacek and Katz [19] demonstrate that nonstandard hulls and

Loeb measures can be constructed in internal-style nonstandardset theories such as BST and IST

4A more detailed analysis of Connesrsquos views can be found in Sanders ([33] 2020)and references therein


bull The theory BST can seamlessly be extended to HST a non-standard set theory that axiomatizes also external sets (seeKanovei and Reeken [24]) In HST the constructions of non-standard hulls and Loeb measures can be carried out in waysanalogous to those familiar from the model-theoretic approach

9 Conclusion

In this paper we establish that infinitesimal methods in ordinarymathematics require no Axiom of Choice at all or only those weakforms of AC that are routinely used in the traditional treatments Thisconclusion follows from the fact that the theory SPOT and its vari-ous strengthenings which do not imply the existence of nonprincipalultrafilters over N or other strong forms of AC are sufficient to carryout infinitesimal arguments in ordinary mathematics (and beyond)

But most users of nonstandard analysis work with hyperreals andthe existence of hyperreals does imply the existence of nonprincipalultrafilters over N So it would seem that ultrafilters are needed afterall However this view implicitly assumes that set theory like ZFCbased exclusively on the membership predicate isin is the only correctframework for the Calculus

Historically5 the Calculus of Newton and Leibniz was first made rig-orous by Dedekind Weierstrass and Cantor in the 19th century usingthe ε-δ approach It was eventually axiomatized in the isin-language asZFC After Robinsonrsquos development of nonstandard analysis it wasrealized that Calculus with infinitesimals also admits a rigorous for-mulation closer to the ideas of Leibniz Bernoulli Euler (see [1]) andCauchy (see [3]) It can be axiomatized in a set theory using the st-isin-language The primitive predicate st can be thought of as a formaliza-tion of the Leibnizian distinction between assignable and inassignablequantities Such theories are obtained from ZFC by adding suitableversions of Transfer Idealization and Standardization

Now that it has been established that the infinitesimal methods donot carry a heavier foundational burden than their traditional counter-parts one can ask the following question Which foundational frame-work constitutes a more faithful formalization of the techniques of the17ndash19 century masters For all the achievements of Cantor Dedekindand Weierstrass in streamlining analysis built into the transformationthey effected was a failure to provide a theory of infinitesimals whichwere the bread and butter of 17ndash19 century analysis until Weierstrass

5See for example Katz and Sherry [25] and Bair et al [2]


By the yardstick of success in formalization of classical analysis ar-guably SPOT SCOT and other theories developed in the presenttext are more successful than ZF and ZF + ADC

One can learn to work in the universe of an st-isin-set theory intu-itively This universe can be viewed as an extension of the standardset-theoretic universe either by a (soritical) predicate st (the internalpicture) or by new ideal objects (the standard picture) see Fletcher etal [9] Sec 55 for a detailed discussion6 This extended universe hasa unique set of real numbers constructed in the usual way and con-taining both standard and nonstandard elements It also of course haschoice functions and ultrafilters over N just as the universe of ZFC

does Mathematicians concerned about AC can analyze their methodsof proof and determine whether a particular result can be carried outin one of the theories considered in this paper For most if not all ofordinary mathematics both traditional and infinitesimal the answeris likely to be affirmative It then follows from Theorems A - D thatthese results are just as effective as those provable in respectively ZF

or ZF + ADC

References

[1] J Bair P B laszczyk R Ely V Henry V Kanovei K Katz M KatzS Kutateladze T McGaffey P Reeder D Schaps D Sherry S ShniderInterpreting the infinitesimal mathematics of Leibniz and Euler Journal forGeneral Philosophy of Science 48 2 (2017) 195ndash238 httpdxdoiorg101007s10838-016-9334-z httpsarxivorgabs160500455

[2] J Bair P B laszczyk R Ely M Katz K Kuhlemann Procedures of Leib-nizian infinitesimal calculus An account in three modern frameworks BritishJournal for the History of Mathematics (2021) httpsdoiorg1010802637545120201851120 httpsarxivorgabs201112628

[3] J Bair P B laszczyk E Fuentes Guillen P Heinig V Kanovei M KatzContinuity between Cauchy and Bolzano Issues of antecedents and priorityBritish Journal for the History of Mathematics 35 3 (2020) 207ndash224 httpsdoiorg1010802637545120201770015 httpsarxivorgabs2005

13259

[4] E Bottazzi V Kanovei M Katz T Mormann D Sherry On mathematical

realism and applicability of hyperreals Mat Stud 51 (2019) 200 - 224[5] B van den Berg E Briseid P Safarik A functional interpretation for non-

standard arithmetic Ann Pure Appl Logic 163 12 (2012) 1962ndash1994

6The internal picture fits well with the multiverse philosophy of Hamkins [11] Oneof his postulates is Well-foundedness Mirage Every universe V appears to be ill-founded from the point of view of some better universe ([11] Sec 9) The internalpicture proposes something stronger but in the same spirit the ill-foundednessis witnessed by a predicate st for which (Visin st) satisfies BST This point iselaborated in [9] Section 73


[6] C C Chang H J Keisler Model Theory third ed Studies in Logic and theFoundations of Math 73 North Holland Publ Amsterdam 1990 649 pp

[7] A Connes Noncommutative Geometry the spectral standpoint 2019 arXive-prints httpsarxivorgabs191010407

[8] A Enayat From bounded arithmetic to second order arithmetic via automor-phisms in A Enayat I Kalantari and M Moniri (Eds) Logic in TehranLecture Notes in Logic vol 26 ASL and AK Peters 2006 httpacademic2

americanedu~enayat

[9] P Fletcher K Hrbacek V Kanovei M Katz C Lobry S Sanders Ap-proaches to analysis with infinitesimals following Robinson Nelson and oth-ers Real Analysis Exchange 42(2) (2017) 193ndash252 httpsarxivorgabs170300425 httpmsupressorgjournalsissueid=50-21D-61F

[10] P Halmos Measure Theory Graduate Texts in Mathematics 18 Springer-Verlag New York 1974 (reprint of the edition published by Van NostrandNew York 1950)

[11] J Hamkins The set-theoretic multiverse Rev Symb Log 5 3 (2012) 416ndash449

[12] D Hilbert On the infinite in J van Heijenoort (Ed) From Frege to GodelA Source Book in Mathematical Logic 1879 - 1931 Harvard University Press1967

[13] D Hilbert P Bernays Grundlagen der Mathematik volume 1 SpringerBerlin 1934


[15] P Howard J E Rubin Consequences of the Axiom of Choice Math Surveysand Monographs 59 Amer Math Society Providence RI 1998 433 pp

[16] K Hrbacek Axiom of Choice in nonstandard set theory J Log Anal 48(2012) 1ndash9 httpsdoi104115jla201248

[17] K Hrbacek Relative set theory Some external issues J Log Anal 28 (2010)1ndash37 httpsdoi104115jla201028

[18] K Hrbacek Nonstandard set theory Amer Math Monthly 86 8 (1979) 659ndash677 httpsdoi1023072321294

[19] K Hrbacek M Katz Nonstandard hulls and Loeb measures in internal settheories in preparation

[20] T Jech The Axiom of Choice North-Holland Publ Amsterdam 1973202 pp

[21] T Jech Set Theory Academic Press New York 1978 621 pp[22] V Kanovei K Katz M Katz T Mormann What makes a theory of infinites-

imals useful A view by Klein and Fraenkel Journal of Humanistic Mathemat-ics 8 1 (2018) 108ndash119 httpscholarshipclaremontedujhmvol8

iss17 and httpsarxivorgabs180201972

[23] V Kanovei M Katz A positive function with vanishing Lebesgue in-tegral in Zermelo-Fraenkel set theory Real Analysis Exchange 42(2)(2017) 385ndash390 httpsarxivorgabs170500493 httpmsupress

orgjournalsissueid=50-21D-61F

[24] V Kanovei M Reeken Nonstandard Analysis Axiomatically Springer-Verlag Berlin Heidelberg New York 2004 408 pp


[25] M Katz D Sherry Leibnizrsquos infinitesimals Their fictionality their modernimplementations and their foes from Berkeley to Russell and beyond Erkennt-nis 78 3 (2013) 571-625 httpdxdoiorg101007s10670-012-9370-yhttpsarxivorgabs12050174

[26] H J Keisler Elementary Calculus An Infinitesimal Approach On-line Edi-tion September 2019 available at httpswwwmathwiscedu~keisler

[27] H J Keisler Foundations of Infinitesimal Calculus On-line Edition 2007available at httpswwwmathwiscedu~keisler

[28] H J Keisler Nonstandard arithmetic and Reverse Mathematics Bull SymbLogic 12 1 (2006) 100ndash125 httpwwwjstororgstable3515886

[29] P Loeb Conversion from nonstandard to standard measure spaces and appli-cations in probability theory Trans Amer Math Soc 211 (1975) 113ndash122httpsdoiorg101090S0002-9947-1975-0390154-8

[30] E Nelson Radically Elementary Probability Theory Annals of MathematicsStudies 117 Princeton University Press Princeton NJ 1987 98 pp

[31] A Robinson Non-standard Analysis Studies in Logic and the Foundations ofMathematics North-Holland Amsterdam 1966

[32] S Sanders The unreasonable effectiveness of Nonstandard Analysis Journal ofLogic and Computation 30 1 (2020) 459ndash524 httpsdoiorg101093logcomexaa019 httparxivorgabs150807434

[33] S Sanders Reverse Formalism 16 Synthese 197 2 (2020) 497ndash544 httpdoiorg101007s11229-017-1322-2 httpsarxivorg

abs170105066

[34] W Sierpinski Fonctions additives non completement additives et fonctionsnon mesurables Fundamenta Mathematicae 30 1938 96ndash99

[35] S G Simpson Subsystems of Second Order Arithmetic second ed CambridgeUniversity Press New York 2009 444 pp

[36] R M Solovay A model of set-theory in which every set of reals is Lebesguemeasurable Annals of Mathematics Second Series 92 (1) 1ndash56 httpsdoiorg1023071970696 JSTOR 1970696 MR 0265151

[37] R Sommer P Suppes Finite Models of Elementary Recursive NonstandardAnalysis Notas de la Sociedad Matematica de Chile 15 (1996) 73-95

[38] M Spector Extended ultrapowers and the VopenkandashHrbacek theorem withoutchoice Journal of Symboic Logic 56 2 (1991) 592ndash607 httpsdoiorg1023072274701

[39] M Spector Iterated extended ultrapowers and supercompactness withoutchoice Annals Pure Applied Logic 54 (1991) 179ndash194 httpsdoiorg1010160168-0072(91)90030-P

[40] K Stroyan Foundations of Infinitesimal Calculus available at http

homepagedivmsuiowaedu~stroyanInfsmlCalculusFoundationsTOC

htm

[41] P Vopenka Calculus Infinitesimalis pars prima 2nd Edition Kanina OPSPilsen 2010 154 pp

[42] Zach R Hilbertrsquos program then and now in Philosophy of Logic Handbookof the Philosophy of Science 2007 pp 411ndash447


Department of Mathematics City College of CUNY New York NY

10031

Email address khrbacekicloudcom

Department of Mathematics Bar Ilan University Ramat Gan 5290002

Israel

Email address katzmikmathbiuacil

1 Introduction

11 Axiom of Choice in Mathematics

12 Countering the objection

13 SPOT and SCOT


21 Some consequences of SPOT

22 Mathematics in SPOT



41 Forcing according to Enayat and Spector

42 Extended ultrapowers



7 Idealization

71 Idealization over uncountable sets

72 Further theories

8 Final Remarks

81 Open problems

82 Forcing with filters

83 Zermelo set theory

84 Weaker theories

85 Finitistic proofs

86 SPOT and CH

87 External sets

9 Conclusion

References


42 Extended ultrapowers 215 Conservativity of SCOT over ZFc 246 Standardization for parameter-free formulas 287 Idealization 3171 Idealization over uncountable sets 3272 Further theories 348 Final Remarks 3681 Open problems 3682 Forcing with filters 3783 Zermelo set theory 3784 Weaker theories 3785 Finitistic proofs 3886 SPOT and CH 3887 External sets 399 Conclusion 40References 41

1 Introduction

Many branches of mathematics exploit the Axiom of Choice (AC)to one extent or another It is of considerable interest to gauge howmuch the Axiom of Choice can be weakened in the foundations ofnonstandard analysis Critics of analysis with infinitesimals often claimthat nonstandard methods require more substantial use of AC thantheir standard counterparts The goal of this paper is to refute such aclaim

11 Axiom of Choice in Mathematics We begin by consider-ing the extent to which AC is needed in traditional non-infinitesimalmathematics Simpson [35] introduces a useful distinction between set-theoretic mathematics and ordinary or non-set-theoretic mathematicsThe former includes such disciplines as general topology abstract al-gebra and functional analysis It is well known that fundamental the-orems in these areas require strong versions of AC Thus

bull Tychonoffrsquos Theorem in general topology is equivalent to fullAC (over Zermelo-Fraenkel set theory ZF)bull Prime Ideal Theorem asserts that every ring with unit has a

(two-sided) prime ideal PIT is an essential result in abstractalgebra and is ldquoalmostrdquo as strong as AC (it is equivalent overZF to Tychonoffrsquos Theorem for Hausdorff spaces) It is also


















































full


SCstandardparams

SP SC full



































n


nand in the second













Qst

1 x1 Qst






1 x1 Qst






1 x1 Qst





holds for all























(h 6= 0rarr

F (c+ h)minus F (c)

h= d+ k

)














(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)



af(t) dt = r iff




(Σib


)



ibi=ia



af(x) dx = r iff


(Σib


)iff


(Σib


)






( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References


















































full


SCstandardparams

SP SC full



































n


nand in the second













Qst

1 x1 Qst






1 x1 Qst






1 x1 Qst





holds for all























(h 6= 0rarr

F (c+ h)minus F (c)

h= d+ k

)














(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)



af(t) dt = r iff




(Σib


)



ibi=ia



af(x) dx = r iff


(Σib


)iff


(Σib


)






( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References























n


nand in the second













Qst

1 x1 Qst






1 x1 Qst






1 x1 Qst





holds for all























(h 6= 0rarr

F (c+ h)minus F (c)

h= d+ k

)














(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)



af(t) dt = r iff




(Σib


)



ibi=ia



af(x) dx = r iff


(Σib


)iff


(Σib


)






( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References





1 x1 Qst





holds for all























(h 6= 0rarr

F (c+ h)minus F (c)

h= d+ k

)














(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)



af(t) dt = r iff




(Σib


)



ibi=ia



af(x) dx = r iff


(Σib


)iff


(Σib


)






( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References












(h 6= 0rarr

F (c+ h)minus F (c)

h= d+ k

)














(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)



af(t) dt = r iff




(Σib


)



ibi=ia



af(x) dx = r iff


(Σib


)iff


(Σib


)






( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References







(2)

int b

a

f(t) dt = sh(Σib

i=iaf(ti) middot h

)



af(t) dt = r iff




(Σib


)



ibi=ia



af(x) dx = r iff


(Σib


)iff


(Σib


)






( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References




( z V = h gt 0rarr


= k

)














The principle CCst



R

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References

































Let E =⋃infin



⋃ν



n=0rn + ε





n=0m(En)































Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References


























Gns) for






)



isin xn2)








)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References






)(9) 〈p q〉 13 (φ and ψ)(Gn1

Gns) iff







Gm)









)









xns)



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References



xns) and

φ(xn1 xns










xns xm)

Hence


xns v)



xns v)












Gns)























p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References




















p extends p2rdquo




analogously













(1) N φ(Gn1 Gns

)



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References



)holds in M


xns)








Gns) by the induc-



Gns) then N ψ(Gn1

Gns)



notψ(Gn1 Gns


If N φ(Gn1 Gns

) then N ψ(Gn1 Gns





Gns w) then




ψ(Gn1 Gns


)


into N















































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References











































⋃infin












































xns〉







)



Case (8)


)






prime1 σ1









v)


σ2[q2] =p π

k1σ1



〈xn1 xns

〉 = πkprime2



prime2


πkprime1σ1 [q

prime1] =pprime π










Gm)






2minus1〉 isin (π

kprime2


〈xn1 xns


σ1n(qprimeprime1)










Gns)
















= emptyk otherwise




iisinp1s(i)

)cap(⋃

iisinp2s(i)

)= empty










7 Idealization






























































r hence θ holds





















j=0 Cj then





















8 Final Remarks

81 Open problems














































9 Conclusion










or ZF + ADC

References




13259









americanedu~enayat





























abs170105066









htm





10031



Israel


1 Introduction



13 SPOT and SCOT










7 Idealization


72 Further theories

8 Final Remarks

81 Open problems



84 Weaker theories


86 SPOT and CH

87 External sets

9 Conclusion

References

arxiv:2009.04980v2 [math.lo] 14 feb 2021

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