saharon shelah- nice infinitary logics

8/3/2019 Saharon Shelah- Nice Infinitary Logics

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NICE INFINITARY LOGICS

SH797

SAHARON SHELAH

Abstract. We deal with soft model theory of infinitary logics. We find alogic between L,0 and L, which has some striking properties. First, ithas interpolations (it was known that each of those logics fail interpolationthough the pair has). Second, well ordering is not characterized in a strongway. Third, it can be characterized as the maximal such nice logic (in fact, isthe maximal logic stronger than L,0 and which satisfies well ordering isnot characterized in a strong way).

Anotated Content

0 Introduction for non-logicians, pg.2

1 Introduction and Preliminaries, pg.5

2 Introducing the logic L1, pg.15

3 Serious properties ofL, pg.22

4 L1 is strong and sum/product of theories, pg.32

Date: May 27, 2011.1991 Mathematics Subject Classification. [2010] Primary 03C95; Secondary: 03C80, 03C55.Key words and phrases. model theory, soft model theory, characterization theorems, Lindstrom

theorem, interpolation, well ordering.The author thanks Alice Leonhardt for the beautiful typing. The author thanks the Israel

Science Foundation for partial support of this research. Part of this work was done while theauthor was visiting Mittag-Leffler Institut, Djursholm, Sweden, in Fall 2000 and Fall 2009. Wethank the Institut for hospitality and support. First handwritten version, Oct.2000.Publication No. 797.

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NICE INFINITARY LOGICS SH797 3

(b) if is a vocabulary, M is a -model, A |M|, |A|+||+0 < Mthen there is a -model N of cardinality , a sub-model of M such

that A |N| and ThL(N) = ThL(M) where ThL(M) = { L(M) : a sentence M |= }.

This means that first order logic does not distinguish infinite cardinals.

2 compactness: if T is a set of sentences in L() and every finite T T has

a model, i.e. for some -model M we have T M |= then T has amodel.

The desirability of this should be obvious.

3 interpolation: if 0 = 1 2 are vocabularies, 1 L(1), 2 L(2)and 1 2, i.e. there is no model of 1 2, (equivalently if M is a(1 2)-model and M |= 1 then M |= ), then there is L(0) such

that 1 and 2.First order logic satisfies interpolation: this is Craig theorem. Lindstrom set outto show that first order logic is the natural choice, recalling there are many logics;for this he has first to define a logic, essentially (see more in Definition 1.9)

4 a logic L consists of the following

(a) a set of sentences L() for any vocabulary , we can define formulas(x0, . . . , xn1) by adding to individual constants

(b) satisfaction relation |=L, i.e. M |=L where M a model, L(M)

(c) natural properties like preservation under isomorphisms and mono-tonicity (i.e. 1 2 L(1) L(2)).

This seems too wide so (see more in Definition 1.12).

5 L is a nice logic if the sets of sentences L() has some natural closureproperties like:

if 1, 2 L() then for some L() we have:M |= iff M |= 1 and M |= 2.

There is a natural order on the class of logics:

6 L1 L2 iff for every vocabulary and 2 L1() there is 2 L2()such that: if M is a -model then M |=L1 1 iff M |=L2 2

7 L1,L2 are equivalent ifL1 L2 and L2 L1.

Now we can phrase (really just one of the versions1

of) (Lindstrom theorem) The logic L is equivalent to L, first order logic when

(a) L is a nice logic

(b) L satisfies LST to 0: i.e. 1(a)

(c) L satisfies compactness, see 2

1e.g. compactness just for countable theorems but then we have to add the occurance numberis 0 or just 0, see Definition 1.11.


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1. Introduction and preliminaries

1(A). Aims.

We feel that this is an important one among my work and will attract littleattention. Is this an oxymoron? We do not think so. See below.

The investigation of model theoretic logics and soft model theory has startedwith Lindstrom theorems and it was a central topic of model theory in the seven-ties. Major aims were to find characterization theorems, new important logics andnon-trivial implications. The achievements were to a large extent summed up inthe handbook Barwise-Feferman [Be85] but then the subject become quite muted.There were some external reasons: stability theory and theoretical computer sci-ence draw people away and there were also some incidental personal reasons. Butprobably the profound reason was a disappointment. The impression was that therewere just too many examples and counterexamples but not enough deep results and,particularly, too many logics and too few characterization theorems (saying a logicL is the unique logic such that ...). Recall that Lindstrom characterize first orderlogic; e.g. as the only reasonable logic satisfying compactness (for 0 sentences)and the downward LST theorem (for one sentence, to 0), (but see (1D) below).Still there was some activity later, particularly of Vaananen.

Here we try to reopen the case. A property which remains mysterious was inter-polation, see Makowsky [Mak85] in the handbook. It was known that L1,0 has in-terpolation, (Lopez-Escobar) but not L, when (, ) = (0, 0), (1, 0),(Malitz).On L, see Dickman [Dic85]. However, the pair (L L werediscovered ([Sh:18]).

The introduction of [Be85] mentions the (then latest advance): some compactlogic strengthening first order logic satisfies the Beth definability theorem, ([Sh:199])a puzzling result. Also the pair (L(Qcf0),L(aa)) of logics satisfies interpolation.Again a puzzling result. Those cases give place to hope of better results usingrelated new logics. Returning to infinitary logics, old problems are (and will be ourmain concern):

{z0}Problem 1.3. Is there a logic L satisfying interpolation such that L


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6 SAHARON SHELAH

{z1}Problem 1.4. Assume is strong limit singular of cofinality 0. Is there a logic

L satisfying interpolation such that L+

,0 L L+

,?Later we have asked ourselves:{z2}

Problem 1.5. Is there, for arbitrarily large cardinal , a logic L such that:

(a) L, L and has reasonable closure properties

(b) L has the downward LST property in the sense that every sentence whichhas a model N, has a model N of small cardinality, moreover M N

(c) L has interpolation

(d) undefinability of well ordering (in a strong sense) which means: if M ex-pands (H(), ) and M |= then for some N we have() N |=

() ordN is not well founded

a posteriori we add() |N| is the union of 0, an internal set of bounded cardinality, i.e. for

some an : n < , , N |= a cardinal such that |an| and(b N)(

n


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() well ordering is not an expressible3

() addition of theories, see 3.14

() product of (two) theories, see 3.14(F) alternative characterization: L1 is a minimal nice logic L for which

for any ordinal < we can characterize (,


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8 SAHARON SHELAH

there are many relatives of similar good properties, without aspecial reason to prefer one or another.

How good is a characterization theorem? Of course, it all depends on the propertiesappearing in the characterization being natural and preferably well established.

The situation of having reasonable logics which we can strengthen presevingtheir main positive properties but neither seeing a maximal one nor proving suchextension do not exist has been prominent in the area; e.g. the most well establishedones like L(Q) = L(1) and also L+,0 .

Why here we tend to look at strong limit cardinals, in particular, = ? Notethat in first order formula with a fix finite vocabulary, with predicates only fortransparency the number of sentences of quantifier depth q has order of magnitudeq, iterated power q times. In infinitary logic, if we like that still there is a sentenceexpressing the quantifier depth theory we need = . A price is that for singular, we lose full substitution, and moreover, full closure under conjunctions

of < . This is resolved if we demand is strong limit regular, i.e. (strongly)inaccessible, fine but the existence of such cardinals is unprovable in ZFC.

1(D). Directions ignored here. We do not deal here with some other majordirections:

3 (a) 0-compact (nice logic)

(b) logics without negation and continuous logic

(c) almost isomorphism (and absolute logic).

We may look at model theory essentially replacing isomorphic by almost iso-morphic, that is isomorphisms by potential isomorphisms, i.e. isomorphism insome forcing extension. In [Sh:12] we have suggested to reconsider a ma jor themein model theory, counting the number of isomorphism types. We call M, N almost-isomorphic when M, N have (the same vocabulary and) the same L,0 -theory,equivalently isomorphic in some generic extension. For a theory T let Iai(, T)be |{M/ L,0 : M a model of T of cardinality }|. This behaves nicely: if T

has cardinality , is first order or just L+,0 then Iai(, ) <

Iai(, T) Iai(, T), (on Iai(, T) for 0-stable T, see a work of Laskowski-Shelahin preparation). In [Sh:12] we also define M is ai-rigid, i.e. a = b M (M, a) L,0 (N, a) and have downward LST theorem for it. Later Nadelsuggested further to consider homomorphisms, in particular for abelian groups,see [EM02, Ch.IV,3,pg.487], more Gobel-Shelah [GbSh:880], Gobel-Herden-Shelah[GbHeSh:948]. Barwise characterized the relevant logic, L,0 by absoluteness:(among logics with satisfaction being absolute under forcing it is maximal).

1(E). Can soft model theory be applied?

What about applications of soft model theory? There are some applicationsusing compact logics. See [Sh:384] on extending first order logic by second orderquantifiers restricted in some ways. Of course, the expressive power of the logicdepend on the restriction, see the example below. In cofinality logic, L(QcfC) withC is a class of regular cardinals, we are allowed to say: the formula (x, y), possiblywith parameters, define a linear order with no last element of cofinality from C,


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recalling that the cofinality of a linear oder I is the minimal cardinality of anunbounded (equivalently cofinal) subset. If C is non-trivial, this logic is a very

interesting logic (e.g. fully compact), in particular showing what we cannot prove sofull compactness is not sufficient to characterize first order logic. But its expressivepower is weak so we do not expect it to have applications.

In [Sh:384] (where you can find something about the history of the topic) weprove the compactness of the quantifier (that is for first order logic extended byit) Lceab = L(Qceab) quantifying over complete embeddings of one atomlessBoolean ring into another. Moreover, for this logic we prove completeness for anatural set of axioms. Now consider the problem can the automorphism groupsof a 1-homogeneous4 Boolean algebra be non-simple5? Much is known on thisgroup and, in particular, that it is almost simple see Rubin-Stepanek [Rv89].It was known that there may exist such Boolean Algebras as by [Sh:b, IV] in somegeneric extension, all automorphisms of P()/finite are trivial (i.e. induced bypermutations of Z such that {n Z : n 0 but (n) < 0} is finite) and vanDowen note that the group of trivial automorphisms of P()/finite is not simple(as the subgoup of the automorphisms induced by permutations of is a normalsubgroup) and the quotient is isomorphic to (Z, +). Alternatively, Koppelberg[Kop85] has directly constructed such Boolean Algebras of cardinality 1 assuming(the more natural assumption) CH. So by the completeness theorem (as the set ofaxioms is absolute), as the relevant facts are expressible in L(Qceab), the existenceis proved in ZFC. Some may want to consider a direct proof. It almost certainlywill give more specific desirable information.

Another helpful quantifier is on branches of trees (see [Sh:72]). In Fuchs-Shelah[FuSh:766] it is used to eliminate the use of diamond, i.e. to prove in ZFC theexistence of valuation domains R such that there are R-modules which are univerasl(i.e. the family of sub-modules is linearly ordered by inclusion) but not standard.

Note the obvious examples (which are called standard): R itself or appropriatequotients. Actually the completeness theorem for this logic gives an absolutenessresult which is used.

We believe that quantifiers with completeness and compactness will be useful soit is worthwhile to find such quantifiers. Hopefully see more in [Sh:800].

1(F). Preliminaries.

{z10}Notation 1.8. 1) denotes a vocabulary, i.e. a set of predicates so each P hasarity(P) < places and each function symbol F has arity(F) < places;

of course, individual constants are zero-place function symbols and we may writearity(P), arity(F) when is clear from the context.1A) For a structure M let M be the vocabulary of M, for a predicate P fromM, PM is the interpretation ofP so an arity(P)-place relation on |M|, the universeof M; similarly for a function symbol F from and in particular for an individualconstant c from .

4A Boolean algebra B is 1-homogeneous if it is atomless for every a, b B \ {0B} we haveB = B b (equivalently for a, b B \ {0B , 1B} for some automorphism f ofB, f(a) = b)

5That is has no normal subgroup which is neither the full group nor the one-element sub-group


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10 SAHARON SHELAH

2) L denotes a logic, see Definition 1.9.3) x, y, z denote sequence of variables (with no repetition). Usually x = xi : i <

so = g() but even possibly x = xs : s S and then we let g(x) = S.4) We say is a relational vocabulary when it has no function symbol.

Recall{z21}

Definition 1.9. 1) A logic L consists of

(a) function L() giving a set of sentences (or formulas (x), see1.10 below) for any vocabulary ; the function is a class function that is adefinition

(b) |=L, satisfaction, i.e. the relation M |=L for M a model, L(M)

(c) renaming: the function , depending on (1, 2, ,L), is a one-to-one func-tion from L(1) onto L(2) when is an isomorphism from the vocabulary

1 onto the vocabulary 2 (i.e. ifP 1 then (P) 2 is a predicateand arity1(P) = arity2((P)) and similarly for F 1)

(d) if is an isomorphism from the vocabulary 1 onto the vocabulary 2 andM1 is a 1-model and M2 = (M1) naturally defined then L(1(M1)) [M1 |= M2 |= ()]

(e) (isomorphism): if M1, M2 are isomorphic -models and L() thenM1 |=L M2 |=L

(f) (monotonicity): if 1 2 then L(1) L(2) and for any 2-model M2and L(1) we have M2 |= (M21) |= .

{z24}Convention 1.10. We define a formula = (x) in L() as a sentence in L({ci : i < g(x)}) with ci(i < ) pairwise distinct individual constants not from

and if(x) L() and a g(x)

M then M |=L [a] means M+

|=L where M+

is the expansion of M by cM+

i = ai for i < g(x), in fact, g(x) can be any indexset.

{z26}

Definition 1.11. For a logic L, the occurance number oc(L) ofL is the minimalcardinal (or ) such that L() = {L() : is of cardinality < } forany vocabulary .

{z29}Definition 1.12. We say a logic L is nice when:

(a) () applying predicates: ifP is an n-place predicate thenP(x0, . . . , xn1) L() is a formula ofL()

() equality: x0 = x1 L(), i.e. for any individual constants c0, c1

there is L

() such that M |= iff cM

0 = cM

1() applying6 functions: for an n-place function symbol F ,

x0 = F(x1, . . . , xn1) is a formula ofL()

(b) () L is closed under conjunction, i.e. for every 1, 2 L() there is3 L() such that for every -model M we have:M |= 3 iff M |= 1 and M |= 2

6We may put together clauses (a)() and (a)() allowing p(0(x), . . . , n1(x)), a term; thechoice is immaterial.


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() L is closed under existential quantifier, (x), i.e. for any L({c}), c an individual constant not in , there is L() such

that for any -model M we have M |= iff M+

|= for some( {c})-expansion M+ of

() L is closed under negation, i.e. for any L() there is L()such that for any -model M, ML M |=L

(c) restricting a sentence to a predicate P, P L() when L(),where P is a unary predicate, and is a relational vocabulary, seeDefinition 1.15, 1.8(4), (but see clause (d)() below)

(d) weak substitution, that is, has substitution for very simple schemes, see1.17 below.

{z31}

Remark 1.13. 1) Above we prefer (a)() on using R(0, . . . , m1) where each = (x) is a term.2) Below we can define the multi-sort vesrion.

{z32}Definition 1.14. 1) A logic L satisfies interpolation when for any sentence 1 L(1), 2 L(2) and = 1 , we have: if 1 2, i.e. for any (1 2)-modelM, M |= 1 M = 2, then for some sentence L() we have 1 and 2.2) Naturally definition for multi-sort languages.

{z34}Definition 1.15. 1) We say P where , L() or pedantically7 =P where P is a unary predicate in the vocabulary , is of minimal cardinalitysuch that L() when:

for any -models M, M |= (P) iff PM is non-empty, closed under FM for

every function symbol from and MPM |= , see below.2) For a -model M and unary predicate P let N = M be the -model with

universe PM, for n-place predicate Q we have QN = QM n|N| and forn-place function symbol F , FN(a) = b a n|N| b N FM(a) = b.

{z36}Definition 1.16. Let L be a logic.0) atL() is the set of atomic formulas (x) from 1.12(a); bsL() = {, : atL(); we may omit L if clear from the context.

1) Let L where , L() mean that M |= M |= for any -model(this does not depend on by Definition 1.9).2) We say is an (L, 1, 2)-interpretation scheme when:

(a) L a logic

(b) 1, 2 vocabularies

(c) = (x)(x) : (x) atL(2)

(d) (x)(x) is a formula in L(1) so = P(x0, . . . , xn1), x = x : < n or = (x0 = F(x1, . . . , xn)), x = x : n.

2A) Above we say the ((L1, 1, 2)-interpretation) scheme is simple when x0=x1has the form (x0 = x1).

7of course is not uniquely determined. We may like to be more liberal in restricting a

model, in 1.8(1) allow FM to be a partial function for F a function symbol from (M). We maycombine this with restriction (see Definition 1.15), then we still have to demand PM = (exceptif we go further and allow empty models). Note M |= F(a) = b when FM(a) is not welldefined.


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3) For as above we say N = N[M] = M1[] when

(a) |N| = |M|/=(x0, x1) which means

EM= = {(a, b) : M |= (a, b)} is an equivalence relation on |M| or justsome non-empty subset

|N| = {a/EM= : a M and M |= =[a, a]}(b) N |= [a] M |= (x)[a] for (x) at(2) and a

>M of length g(x)

(note: not for every such and 1-model M1 is N[M2] well defined, we need that=(, ) defines an equivalence relation on |M1|, which is a congruence relation forthe 2-relations and functions we define and, of course, the definition of functionsgives ones, similarly below; but if is simple this problem does not arise).4) We say the logic L satisfies full substitution8 when: if 1, 2 are vocabularies, = (x)(x) : (x) atL(2) is a simple (L, 1, 2)-interpretation scheme, and2 L(2), then there is 1 L(1) such that: if M1 = N[M2], see below, so

M is a -models for = 1, 2 then M1 |= 1 M2 |= 2.5) We say that the logic L satisfies substitution when we require to be simple.

{z37}Definition 1.17. 1) We say is a weak (L, 1, 2)-scheme (but L is immaterialso can be omitted) when :

(a) 1, 2 are vocabularies

(b) = (x)(x) : (x) at(2)

(c) the formulas (x) are atomic (on conjunction of two atomic for equality)formulas in which we substitute some variables by individual constants,moreover:

() if (x) is equal to (x0 = x1) so g(x) = 2 then (x) = (x0 = x1 P(x)(x0, c(x))

() if (x) = P(x), so P 2 a predicate then

1 (x) = Q(x)(x, c) where

2 Q(x) a predicate from 13 c a sequence of individual constants from 14 so arity2(P) = g(x), arity1(Qp) = g(x) + g(c)

() if (x) = (x0 = F(x1, . . . , xn)) so F 2 a function symbol then

1 (x)(x) is (x0 = HF(x1, . . . , xn, cF)) or QF(x0, . . . , xn, cF)

2, 3 as above.

2) We say L has weak substitution when as in Definition 1.16(4), but restrictingourselves to weak schemes.

{z38}

Remark 1.18. 1) The meaning of 1.17(1)(c)() is

()1 (x)(x) = (x0 = HF(x1, . . . , xn, cF))

HF 1 a function symbol of arity n + g(cF)

cF a sequence of individual constants fromor

()2 similarly using QF(x0, . . . , xn, cF).

8We do not use full substitution; as for singular L0,L1 are not closed under full substitution


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2) Why in 1.17(1)(c) the moreover, i.e. why is the second version stronger? Aswe demand the cs to be as in the end. Does not matter whether we ask it or not.

We could further demand c = c for all at(2).3) In weak (L, 1, 2)-scheme (hence in weak substitution) we may use firstorder formulas (instead of atomic), i.e. that is our results will not be affected bythis change in the definition.4) Also, in weak(L, 1, 2)-scheme we may add c = c, for one c, and/ordemand is simple. For the later change in the proof of 3.4, we have to say thatwithout loss of generality all M,n have cardinality 1 and in fact = 1, forthis we need claim version of LST, see so have to add the assumption L satisfiessuitable version of LST.

{z39}

Definition 1.19. 1) For logics L1,L2 let L1 L2 mean that: L1() L2()for any vocabulary and M |=L1 is equivalent to M |=L2 when L() andM is a -model.

2) For logics L1,L2 let L1 L2 or L2 is stronger than L2 means that forevery and L1() there is L2() such that , are equivalent, i.e.M |=L1 M |=L2 for any -model M.3) We say that the logics L1,L2 are equivalent, L1 L2 when L1 L2 andL2 L1.

{z42}Definition 1.20. 1) The logic L, for 0 is defined like first order logic,but L,() is the closure of the set of atomic formulas under ,

i


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seems pointless to allow to be singular as was originally done. Still this is thetradition and we use it.

2) If is a regular cardinal then the logics L0, andL

0


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2. The logic L1

Our main definition is{a8}

Definition 2.1. For a vocabulary , -models M1, M2, set formulas in the vocab-ulary in any logic (each with finitely many free variables if not said otherwise (see2.3(4)), cardinal and ordinal we define a game = ,,[M1, M2] as follows,and using (M1, b1), (M2, b2) with their natural meaning when Dom(b1) = Dom(b2).

(A) The moves are indexed by n < (but every actual play is finite), justbefore the n-th move we have a state sn = (A

1n, A

2n, h

1n, h

2n, gn, n, n)

(B) s = (A1, A2, h1, h2, g , , n) = (A1s , A2s , h

1s , h

2s , gs, s, ns) is a state (or n-state

or (, n)-state or (,< )-state) when:(a) A [M] for = 1, 2

(b) so an ordinal(c) h is a function from A into

(d) g is a partial one-to-one function from M1 to M2 and let g1s = g1 =

gs = g and let g2s = g2 = (g1s)

1,

(e) Dom(g) A for = 1, 2

(f) g preserves satisfaction of the formulas in and their negations, i.e.for (x) and a g(x)Dom(g) we have M1 |= [a] M2 |= [g(a)]

(g) if a Dom(g) then h(a) < n(C) we define the state s = s0 = s0 by letting ns = 0, A

1s = = A

1s , s =

, h1s = = h2s , gs = ; so really s depend only on (but in general, this

may not be a state for our game as possibly for some sentence wehave M1 |= M2 |= )

(D) we say that a state t extends a state s when As At, h

s h

t for = 1, 2

and gs gt, s > t, ns < nt; we say t is a successor of s if in additionnt = ns + 1

(E) in the n-th movethe anti-isomorphism player (AIS) chooses (n+1, n, A

n) such that:

n {1, 2}, n+1 < n and Ann A

n [Mn]

,the isomorphism player (ISO) chooses a state sn+1 such that sn+1 is a successor of sn

Ansn+1 = An

A3nsn+1 = A3nsn Dom(g3nsn+1 ) if a An\A

nsn

then hnsn+1(a) n + 1

Dom(gnsn+1) = {a Ansn

: hnsn(a) < n + 1} so it includes Dom(gnsn

)

sn+1 = n+1.(F) the play ends when one of the player has no legal moves (always occur

as n < n1) and then this player loses, this may occur for n = 0

for = 0 we stipulate that ISO wins iff s0 is a state.


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{a9}Discussion 2.2. 1) This is a parallel to EF-games. Note that we like on the one

hand the game to have moves, really each play has < moves and deal withsets of cardinality and on the other hand we do not like well ordering to bedefinable, i.e. allow M1 to be well ordered while M2 to be non-well ordered but stillthe ISO player wins. We do this by rescheduling our debts, i.e. using the hns.{a10}Definition 2.3. 1) Let E0,,, be the class {(M1, M2) : M1, M2 are -models andin the game ,,[M1, M2] the ISO player has a winning strategy} where is a setof formulas in the vocabulary , each with finitely many free variables.2) E1,,, is the closure ofE

0,,, to an equivalence relation (on the class of -models).

3) Above we may replace by qf() which means = the set at() or bs() formulasin the vocabulary .4) Above if we omit we mean = and if we omit we mean bs(). Abusing

notation we may say M1, M2 are E0,

,,-equivalent.{a12}

Fact 2.4. Assume ,,[M1, M2] is well defined and M1, M2 are -models.1) The game ,,[M1, M2] is a determined game and is without memory, i.e.during a play, being a winning situation does not depend on the history, just onthe current state, also only M are relevant.2a) The relation E0,, holds for (M1, M2) when M1, M2 are isomorphic -models.

2b) If M1 = M1, M2

= M2 then M1E0

,,M2 M1E

0,,M

2.

2c) E0,, is reflexive and symmetric.

3) The relation E1,, is an equivalence relation on the class of -models.

4) If is a limit ordinal then M1E0

,,M2 iff [ < M1E0

,,M2].

5) E1,, has +1(|| + ) equivalence classes.

6) If1 2, 1 2, 1 2 and 1 2 and M1E0,2

2,2,2M2 then M1E

0,1

1,1,1M2.

Proof. 1) Obvious.2a) Let g be an isomorphism from M1 onto M2. Now a winning strategy for theplayer ISO in ,,[M1, M2] is to preserve gsn g.2b) Should be clear.2c) Let us check.

Reflexivity:Follows from part (2a) as M = M.Symmetry:Reading the definition carefully it should be clear.

3),4) Easy, too.5) We prove by induction on that there is an equivalence relation E2,, on

the class of ()-models such that M1E2,,M2 M1E0

,,M2 and E2

,, has

+1(|| + ) equivalence classes. For = 0, recall clause (F) of Definition 2.1 soeasily M1E0,,M2 means that = {[a] : M |= and , is a sentence} does

not depend on ; clearly E0,, is an equivalence relation with 2|| equivalence

classes and E1,, = E0

,,; let E2

,, = E0

,,. For a limit ordinal use part (4) and

choose E2,, = {E2

,, : < } recalling (|| + ) = {(|| + ) : < }.

Lastly, for = + 1 use the induction hypothesis and (|| + ) = 2(||+).

Alternatively use L+,+,, i.e. use the proof of 2.8, when we replace (|| + )by (||+)+.6) Easy. 2.4


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Obvious by the definition because for any vocabulary of cardinality theequivalence relation E1qf(),, has +1() equivalence classes by 2.4(5).

Clause (c): Restricting to a unary predicate PEasily if is defined using (qf(), , ) and without loss of generality P

then so is P.Clause (d): We prove more: Full substitution:

Assume we are given vocabularies 1, 2 and consider substituting P(x0, . . . , xarity2(P)1)

L1(0) for P 2 treating F(x0, . . . , xarity(F)1) = xarity(F) as an (arity2(F)+1)-

place predicate. Let P be defined by (qf(+arity(P)), P, P) and let 0 = sup{P :P 2}.

We are given L1(1) and we shall find L1(2) which says that if we

substitute P(x0, . . . , xarity2 (P)) instead P(x0, . . .) in for every P 1, we get

(up to equivalence) . Let be defined by (qf(1), 1, 1). Let 2 = 1 + 0 andeasily there is as required defined by (qf(2), 1, 2). 2.7

{a14}Claim 2.8. 1) Let = + .L1 L

0, i.e. every formula ofL

1 is equivalent to, hence can be looked at, as a

formula ofL0.2) L1 L

1.

{a15}Remark 2.9. For many purposes we identify them, i.e. say L1 L

0; this gives

another reasonable version of subformulas.

Proof. 1) We first prove:

1 if s is a state in the game qf(),,[M1, M2] and = s, = (M), a1 =

a1

: < () +>(M1) list Dom(gs) and a2 = a

2

: < () wherea2 = gs(a

1) and M2 |= [a1] M2 |= [a2] for every = (x : .4) If M1E

1qf(),,M2 and M1 and + + 1 then M1

= M2.

Proof. 1) By (2).2) Toward contradiction, assume this fails; and without loss of generality M1E0qf({R}),+,++1M2.

By symmetry, without loss of generality assume M2 |= ,R but M1 |= ,R. Wesimulate a play of the game qf({R}),+,++1(M1, M2) in which the ISO playeruses a (fixed) winning strategy st.


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Proof. 1) Easy (or use the sum theorem (see 3.16) and part (2)).2),3) Easy. 3.9

{a32} Theorem 3.10. Second Characterization TheoremLet = , thenL1 is the minimal logic L (up to equivalence) such that:

(a) L is a nice logic

(b) L1 L, i.e. L+,0 L for <

(c) ,R is equivalent to some sentence inL

(d) L is -closed

(e) L has dullness-elimination.{a32d}

Remark 3.11. Putting together 3.4, 3.10 we get full characterization.

Proof. First L1 satisfies clause (a) by 2.15(2), clause (b) by 2.15, clause (c) by 3.7(1)and clause (d) by Theorem 3.13 and Observation 3.12(1) below, and lastly L1

has

dullness-elimination by Claim 3.9(1) so together L1 satisfies of 3.10. Second, weshall assume L satisfies and L1() and we shall find

L() equivalentto it (it will be 1 below). By L satisfies weak substitution without loss ofgenerality has predicates only.

We deal with and define naturally [P] L1(), P a new unary predicate andlet = {P} so [P] says (M)PM satisfies .

Let , < and let L1() be defined using (qf(), , ), so || < . We let = +1( + ||) and = 2

. Let + = {R1, R}, arity(R1) = 3, arity(R) = 2and R, R1 /

. For i = 0, 1 let K be the class of +-models M such that:

(a) (|M|, RM) |= ,R and PM =

(b) {Bb : b M} is -directed where Bb = {a M : aRMb}

(c) for each b M the set {(a1, a2) : (a1, a2, b) RM1 } is a well orderingof Bb of order type

(d) if b1, b2 M and Bb1 Bb2 PM then

(M)Bb1 L+,+ (M)Bb2 in particular Bb1 is

closed under FM for any function symbolF

(e) if b M, then for some c we have Bc = Bb PM Bc P

M

(f) if b M and Bb P then (M)Bb |= [P] iff = 1

(g) if Q then QM = QMPM.

Now

()1 clauses (b),(c),(d),(e),(g) of can be described by some L(2)+,0(1).

[Why? Note clause (c), so obvious.]

()2 clause (f) can be expressed by a sentence from L(+).

[Why? Let N : < 2 list the -models with universe . For each b Msuch that Bb PM let gb be the unique one-to-one function from Bb onto someordinal b such that

if a1, a2 Bb then (a1, a2, b) RM1 gb(a) < gb(a2).


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Hence for such b there is a unique b such that gb is an isomorphism from (M)Bbonto N. For each there is a formula (x) L+,0(M) such that for every

b M we have M |= [b] iff b is as above and b = .So for every u 2, u(x) =

u(x) belongs to L(2)+,0(M) hence is equiv-

alent to some u(x) L(M). Now ()2 should be clear.]

()3 K is definable by some L(+) for = 1, 2.

[Why? By clause (b) of the assumption, some L(+) is equivalent to where is from ()1. By clause (c) of the assumption some

L(+) is equivalentto ,R. But by clause (a) of the assumption L is a nice logic, hence (see clauses(b)2, (b)3, (c) of Definition 1.12) there is a sentence

L(

+) equivalent to from ()3. By clause (b)2 of Definition 1.12, there is L(

+) equivalent to so we are done.]

()4 for = 1, 2 the classes {(M1)PM1 : M1 K1}, {(M2)PMi , M2 K2}are equal to {M : M a -model of }, {M : M a -model of }, respec-tively.

By L being -closed and having dullness-elimination (see clauses (d),(e) of thetheorems assumption) we are done. 3.10

We can note and recall{a33}

Observation 3.12. 1) If the logic L satisfies interpolation then it is -closed.2) L is -closed and has dullness-elimination iff L is strongly -closed.

3) IfL satisfies interpolation with finitely many sorts (so the interpolant mentionsonly the common sorts) then L is strongly -closed.

4) The logic L satisfies interpolation and dullness-elimination iff L satisfies in-terpolation for finitely many sort models.5) In 3.10 we can replace clauses (d),(e) by

(d) for every < ,L is -closed in { : = }, see Definition 3.8.

Proof. Should be clear. 3.12{a35}

Theorem 3.13. 1) For = the logic L1 satisfies interpolation.2) Also if L1(1) where < , the vocabularies 0 1 have cardinality

and = + , = 2 then we can find a sequence : < (2

)+ of members ofL12 (0) such that:

(a) for < (2)+

(b) if0 = 12 and L1(2) a sentence such that then

for some < (2)+.

Proof. 1) This means that we should prove the existence of L1(0) such that 1 and 2 when we assume:

1 (a) 1 2 = 0

(b) L1() for = 1, 2

(c) 1 2.


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Toward contradiction we assume this fails. Without loss of generality ( < 3) arevocabularies of cardinality < .

Let be such that < , |1| + |2| and L1 for = 1, 2. For

each [, ) as L1(0) is closed under conjunction (and conjunctions) of formulas and negations, clearly

2 for every < we can choose M1, M2 such that

(a) M1 is a 1-model

(b) M2 is a 2-model

(c) M10, M2 0 are E

1qf(0),||,

-equivalent

(d) M1 |= 1(e) M2 |= 2

and continue naturally as in the proof of 3.4.

2) Similarly. 3.13{a36}

Theorem 3.14. If = then L1 satisfies the addition and product theorems,that is (all the models are -models for a fixed vocabulary ; for the sum case weassume has no function symbols (in particular no individual constants; or onlyunary functions) and the relevant models have disjoint universes):

(a) sum: ThL1(M1 + M2) = ThL1(M1) + ThL1(M2) that is,

(a) if ThL1(M) = ThL1(N) for = 1, 2 then ThL1(M1 + M2) = ThL1(N1 +N2)

(b) product - ThL1(M1 M2) = ThL1(M1) ThL1(M2) that is,

(b) if ThL1(M) = ThL1(N) for = 1, 2 then ThL1(M1 M2) = ThL1(N1 N2)

(c) moreover, we can replace ThL1(N) by N/E1

qf(),, when < , < +.

Proof. It suffices to prove clause (c). We prove it for products. For the sums thisis easier.

Clause (c) for product:Clearly it suffices to prove

assume M = M1 M

2 for = 1, 2 are -models and M

1 E

1qf(),,M

2 for

= 1, 2 then M1E1

qf(),,M2.

So let M,k : k k(, ) be such that M1,0 = M1 , M2,k = M2 and M,kE

0qf(),,M,k+1

for k < k(). Let k() = max{k(1), k(2)} and let M,k = M,k() ifk() < k k().

As E0qf(),, is reflexive, still we have M,kE0qf(),M,k+1 for = 1, 2 and k < k().Let Mk = M1,k M2,k for k k() we have to prove M0E1qf(),,Mk(); by the

definition ofE1qf(),, it suffices to prove for each k < k() that

k MkE0qf(),,Mk+1.

So we have to find a winning strategy for the ISO player for the gameqf(),,[Mk, Mk+1].The ISO player restricts itself, in the n-th move to (, n)-states sn for a play of

qf(),,[Mk, Mk+1] satisfying, for some pair (sn,1, sn,2), that for = 1, 2


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4. L1 is strong

Our first aim is:{c2}

Question 4.1. How strong is L1? It is more like L1 or L

0? E.g. upward LST?

Downward LST to which cardinals?So far we have given indication to its being similar to L1 , however we shall

show below that for the LST, L1 is closer to L0 (see 4.8) and fail the theorem on

the theory of an infinite product (which both L0,L1 satisfies). But restricting

ourselves to cardinals = 0 , the situation for the downward LST is similarto the one for L0, see below.

{c3}Definition 4.2. For a relational vocabulary and cardinal of cofinality > 0 wesay K is a ( , , )-class of structures when :

(a) K is a class of -models each of cardinality

(b) K is closed under submodels of cardinality (c) K is closed under isomorphism

(d) K is closed under increasing union of -chains.

2) For ,, K as above let K L0(2)+,+() be a sentence such that for -models

M we have: M |= K iff for any A M of cardinality there is h : A suchthat for every large enough n, Mh1{n} K.

{c3d}Example 4.3. Let Kwo = {(A,


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{c6}Remark 4.7. Recall cov(,,,) = Min{|P| : P []


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[Why? If n = n() then FNnn() is the identity and this means Nn |= dn,m(2)

saharon shelah- nice infinitary logics

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