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  • Set-theoretic universesFrom Wikipedia, the free encyclopedia

  • Contents

    1 Axiom of constructibility 11.1 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    2 Constructible universe 22.1 What is L? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Additional facts about the sets L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 L is a standard inner model of ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 L is absolute and minimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.4.1 L and large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 L can be well-ordered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 L has a reection principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 The generalized continuum hypothesis holds in L . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 Constructible sets are denable from the ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 Relative constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    3 Grothendieck universe 93.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Grothendieck universes and inaccessible cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    4 Gdel operation 124.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    5 Jensen hierarchy 145.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    i

  • ii CONTENTS

    5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Rudimentary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    6 Minimal model (set theory) 166.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    7 Silver machine 177.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2 Silver machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    8 Statements true in L 19

    9 Von Neumann universe 209.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    9.1.1 Finite and low cardinality stages of the hierarchy . . . . . . . . . . . . . . . . . . . . . . . 219.2 Applications and interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    9.2.1 Applications of V as models for set theories . . . . . . . . . . . . . . . . . . . . . . . . . 219.2.2 Interpretation of V as the set of all sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2.3 V and the axiom of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2.4 The existential status of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    9.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    10 Zero sharp 2510.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Statements that imply the existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Statements equivalent to existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.4 Consequences of existence and non-existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.5 Other sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    10.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

  • Chapter 1

    Axiom of constructibility

    The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set isconstructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and theconstructible universe, respectively. The axiom, rst investigated by Kurt Gdel, is inconsistent with the propositionthat zero sharp exists and stronger large cardinal axioms (see List of large cardinal properties). Generalizations ofthis axiom are explored in inner model theory.

    1.1 ImplicationsThe axiom of constructibility implies the axiom of choice over ZermeloFraenkel set theory. It also settles manynatural mathematical questions independent of ZermeloFraenkel set theory with the axiom of choice (ZFC). Forexample, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslins hypoth-esis, and the existence of an analytical (in fact,12 ) non-measurable set of real numbers, all of which are independentof ZFC.The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater orequal to 0#, which includes some relatively small large cardinals. Thus, no cardinal can be 1-Erds in L. WhileL does contain the initial ordinals of those large cardinals (when they exist in a supermodel of L), and they are stillinitial ordinals in L, it excludes the auxiliary structures (e.g. measures) which endow those cardinals with their largecardinal properties.Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as anaxiom for set theory in the same way as the ZFC axioms. Among set theorists of a realist bent, who believe thatthe axiom of constructibility is either true or false, most believe that it is false. This is in part because it seemsunnecessarily restrictive, as it allows only certain subsets of a given set, with no clear reason to believe that theseare all of them. In part it is because the axiom is contradicted by suciently strong large cardinal axioms. This pointof view is especially associated with the Cabal, or the California school as Saharon Shelah would have it.

    1.2 See also Statements true in L

    1.3 References Devlin, Keith (1984). Constructibility. Springer-Verlag. ISBN 3-540-13258-9.

    1.4 External links How many real numbers are there?, Keith Devlin, Mathematical Association of America, June 2001

    1

  • Chapter 2

    Constructible universe

    Gdel universe redirects here. For Kurt Gdels cosmological solution to the Einstein eld equations, see Gdelmetric.

    In mathematics, in set theory, the constructible universe (or Gdels constructible universe), denoted L, is aparticular class of sets that can be described entirely in terms of simpler sets. It was introduced by Kurt Gdel inhis 1938 paper The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.[1] In this,he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice andthe generalized continuum hypothesis are true in the constructible universe. This shows that both propositions areconsistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold insystems in which one or both of the propositions is true, their consistency is an important result.

    2.1 What is L?L can be thought of as being built in stages resembling the von Neumann universe, V. The stages are indexed byordinals. In von Neumanns universe, at a successor stage, one takes V to be the set of all subsets of the previousstage, V. By contrast, in Gdels constructible universe L, one uses only those subsets of the previous stage that are:

    denable by a formula in the formal language of set theory with parameters from the previous stage and with the quantiers interpreted to range over the previous stage.

    By limiting oneself to sets dened only in terms of what has already been constructed, one ensures that the resultingsets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory andcontained in any such model.Dene

    Def(X) :=nfy j y 2 X and (X;2) j= (y; z1; : : : ; zn)g

    and formula rst-order a is z1; : : : ; zn 2 Xo:L is dened by transnite recursion as follows:

    L0 := ?: L+1 := Def(L): If is a limit ordinal, then L :=

    S

  • 2.2. ADDITIONAL FACTS ABOUT THE SETS L 3

    If z is an element of L, then z = {y | y L and y z} Def (L) = L. So L is a subset of L, which is asubset of the power set of L. Consequently, this is a tower of nested transitive sets. But L itself is a proper class.The elements of L are called constructible sets; and L itself is the constructible universe. The "axiom of con-structibility", aka V=L, says that every set (of V) is constructible, i.e. in L.

    2.2 Additional facts about the sets LAn equivalent denition for L is:

    L =[ . On the other hand, V equals L does imply that V equals L if = , for example if is inaccessible.More generally, V equals L implies H equals L for all innite cardinals .If is an innite ordinal then there is a bijection between L and , and the bijection is constructible. So these setsare equinumerous in any model of set theory that includes them.As dened above, Def(X) is the set of subsets of X dened by 0 formulas (that is, formulas of set theory containingonly bounded quantiers) that use as parameters only X and its elements.An alternate denition, due to Gdel, characterizes each L as the intersection of the power set of L with theclosure of L[fLg under a collection of nine explicit functions. This denition makes no reference to denability.All arithmetical subsets of and relations on belong to L (because the arithmetic denition gives one in L).Conversely, any subset of belonging to L is arithmetical (because elements of L can be coded by naturalnumbers in such a way that is denable, i.e., arithmetic). On the other hand, L already contains certain non-arithmetical subsets of , such as the set of (natural numbers coding) true arithmetical statements (this can be denedfrom L so it is in L).All hyperarithmetical subsets of and relations on belong to L!CK1 (where !

    CK1 stands for the Church-Kleene

    ordinal), and conversely any subset of that belongs to L!CK1 is hyperarithmetical.[2]

    2.3 L is a standard inner model of ZFCL is a standard model, i.e. it is a transitive class and it uses the real element relationship, so it is well-founded. L isan inner model, i.e. it contains all the ordinal numbers of V and it has no extra sets beyond those in V, but it mightbe a proper subclass of V. L is a model of ZFC, which means that it satises the following axioms:

    Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.

    (L,) is a substructure of (V,), which is well founded, so L is well founded. In particular, if xL, thenby the transitivity of L, yL. If we use this same y as in V, then it is still disjoint from x because we areusing the same element relation and no new sets were added.

    Axiom of extensionality: Two sets are the same if and only if they have the same elements.

    If x and y are in L and they have the same elements in L, then by Ls transitivity, they have the sameelements (in V). So they are equal (in V and thus in L).

    Axiom of empty set: {} is a set.

  • 4 CHAPTER 2. CONSTRUCTIBLE UNIVERSE

    {} = L0 = {y | yL0 and y=y} L1. So {} L. Since the element relation is the same and no newelements were added, this is the empty set of L.

    Axiom of pairing: If x, y are sets, then {x,y} is a set.

    If xL and yL, then there is some ordinal such that xL and yL. Then {x,y} = {s | sL and(s=x or s=y)} L. Thus {x,y} L and it has the same meaning for L as for V.

    Axiom of union: For any set x there is a set y whose elements are precisely the elements of the elements of x.

    If x L, then its elements are in L and their elements are also in L. So y is a subset of L. y = {s |sL and there exists zx such that sz} L. Thus y L.

    Axiom of innity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.

    From transnite induction, we get that each ordinal L. In particular, L and thus L.

    Axiom of separation: Given any set S and any proposition P(x,z1,...,z), {x|xS and P(x,z1,...,z)} is a set.

    By induction on subformulas of P, one can show that there is an such that L contains S and z1,...,zand (P is true in L if and only if P is true in L (this is called the "reection principle")). So {x | xS andP(x,z1,...,z) holds in L} = {x | xL and xS and P(x,z1,...,z) holds in L} L. Thus the subset isin L.

    Axiom of replacement: Given any set S and anymapping (formally dened as a proposition P(x,y) where P(x,y)and P(x,z) implies y = z), {y | there exists xS such that P(x,y)} is a set.

    Let Q(x,y) be the formula that relativizes P to L, i.e. all quantiers in P are restricted to L. Q is a muchmore complex formula than P, but it is still a nite formula, and since P was a mapping over L, Q mustbe a mapping over V; thus we can apply replacement in V to Q. So {y | yL and there exists xS suchthat P(x,y) holds in L} = {y | there exists xS such that Q(x,y)} is a set in V and a subclass of L. Againusing the axiom of replacement in V, we can show that there must be an such that this set is a subsetof L L. Then one can use the axiom of separation in L to nish showing that it is an element of L.

    Axiom of power set: For any set x there exists a set y, such that the elements of y are precisely the subsets ofx.

    In general, some subsets of a set in L will not be in L. So the whole power set of a set in L will usuallynot be in L. What we need here is to show that the intersection of the power set with L is in L. Usereplacement in V to show that there is an such that the intersection is a subset of L. Then theintersection is {z | zL and z is a subset of x} L. Thus the required set is in L.

    Axiom of choice: Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containingexactly one element from each member of x.

    One can show that there is a denable well-ordering of L which denition works the same way in Litself. So one chooses the least element of each member of x to form y using the axioms of union andseparation in L.

    Notice that the proof that L is a model of ZFC only requires that V be a model of ZF, i.e. we do NOT assume thatthe axiom of choice holds in V.

  • 2.4. L IS ABSOLUTE AND MINIMAL 5

    2.4 L is absolute and minimalIf W is any standard model of ZF sharing the same ordinals as V, then the L dened in W is the same as the L denedin V. In particular, L is the same in W and V, for any ordinal . And the same formulas and parameters in Def (L)produce the same constructible sets in L.Furthermore, since L is a subclass of V and, similarly, L is a subclass of W, L is the smallest class containing all theordinals that is a standard model of ZF. Indeed, L is the intersection of all such classes.If there is a set W in V that is a standard model of ZF, and the ordinal is the set of ordinals that occur in W,then L is the L of W. If there is a set that is a standard model of ZF, then the smallest such set is such a L. Thisset is called the minimal model of ZFC. Using the downward LwenheimSkolem theorem, one can show that theminimal model (if it exists) is a countable set.Of course, any consistent theory must have a model, so even within the minimal model of set theory there are setsthat are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, theydo not use the normal element relation and they are not well founded.Because both the L of L and the V of L are the real L and both the L of L and the V of L are the real L, we getthat V=L is true in L and in any L that is a model of ZF. However, V=L does not hold in any other standard modelof ZF.

    2.4.1 L and large cardinalsSince OnLV, properties of ordinals that depend on the absence of a function or other structure (i.e. 1ZF for-mulas) are preserved when going down from V to L. Hence initial ordinals of cardinals remain initial in L. Regularordinals remain regular in L. Weak limit cardinals become strong limit cardinals in L because the generalized con-tinuum hypothesis holds in L. Weakly inaccessible cardinals become strongly inaccessible. Weakly Mahlo cardinalsbecome strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinalproperties) will be retained in L.However, 0# is false in L even if true in V. So all the large cardinals whose existence implies 0# cease to have thoselarge cardinal properties, but retain the properties weaker than 0# which they also possess. For example, measurablecardinals cease to be measurable but remain Mahlo in L.Interestingly, if 0# holds in V, then there is a closed unbounded class of ordinals that are indiscernible in L.While someof these are not even initial ordinals in V, they have all the large cardinal properties weaker than 0# in L. Furthermore,any strictly increasing class function from the class of indiscernibles to itself can be extended in a unique way to anelementary embedding of L into L. This gives L a nice structure of repeating segments.

    2.5 L can be well-orderedThere are various ways of well-ordering L. Some of these involve the ne structure of L, which was rst describedby Ronald Bjorn Jensen in his 1972 paper entitled The ne structure of the constructible hierarchy. Instead ofexplaining the ne structure, we will give an outline of how L could be well-ordered using only the denition givenabove.Suppose x and y are two dierent sets in L and we wish to determine whether xy. If x rst appears in Land y rst appears in L and is dierent from , then let x

  • 6 CHAPTER 2. CONSTRUCTIBLE UNIVERSE

    that each parameters possible values are ordered according to the restriction of the ordering of L to L, so thisdenition involves transnite recursion on .The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transnite induc-tion. The values of n-tuples of parameters are well-ordered by the product ordering. The formulas with parametersare well-ordered by the ordered sum (by Gdel numbers) of well-orderings. And L is well-ordered by the orderedsum (indexed by ) of the orderings on L.Notice that this well-ordering can be dened within L itself by a formula of set theory with no parameters, only thefree-variables x and y. And this formula gives the same truth value regardless of whether it is evaluated in L, V, orW (some other standard model of ZF with the same ordinals) and we will suppose that the formula is false if eitherx or y is not in L.It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-orderthe proper class V (as we have done here with L) is equivalent to the axiom of global choice, which is more powerfulthan the ordinary axiom of choice because it also covers proper classes of non-empty sets.

    2.6 L has a reection principleProving that the axiom of separation, axiom of replacement, and axiom of choice hold in L requires (at least as shownabove) the use of a reection principle for L. Here we describe such a principle.By mathematical induction on n suchthat for any sentence P(z1,...,z) with z1,...,z in L and containing fewer than n symbols (counting a constant symbolfor an element of L as one symbol) we get that P(z1,...,z) holds in L if and only if it holds in L.

    2.7 The generalized continuum hypothesis holds in LLet S 2 L , and let T be any constructible subset of S. Then there is some with T 2 L+1 , so T = fx 2L : x 2 S ^ (x; zi)g = fx 2 S : (x; zi)g , for some formula and some zi drawn from L . By thedownward LwenheimSkolem theorem, there must be some transitive set K containing L and some wi , andhaving the same rst-order theory as L with the wi substituted for the zi ; and this K will have the same cardinalas L . Since V = L is true in L , it is also true in K, so K = L for some having the same cardinal as . AndT = fx 2 L : x 2 S ^ (x; zi)g = fx 2 L : x 2 S ^ (x;wi)g because L and L have the same theory. SoT is in fact in L+1 .So all the constructible subsets of an innite set S have ranks with (at most) the same cardinal as the rank of S; itfollows that if is the initial ordinal for +, then L\P(S) L+1 serves as the powerset of S within L. And thisin turn means that the power set of S has cardinal at most ||||. Assuming S itself has cardinal , the power setmust then have cardinal exactly +. But this is precisely the generalized continuum hypothesis relativized to L.

    2.8 Constructible sets are denable from the ordinalsThere is a formula of set theory that expresses the idea that X=L. It has only free variables for X and . Using thiswe can expand the denition of each constructible set. If sL, then s = {y|yL and (y,z1,...,z) holds in (L,)}for some formula and some z1,...,z in L. This is equivalent to saying that: for all y, ys if and only if [there existsX such that X=L and yX and (X,y,z1,...,z)] where (X,...) is the result of restricting each quantier in (...) toX. Notice that each zL for some

  • 2.9. RELATIVE CONSTRUCTIBILITY 7

    yield. But the point remains, there is a formula of set theory that is true only for the desired constructible set s andthat contains parameters only for ordinals.

    2.9 Relative constructibilitySometimes it is desirable to nd a model of set theory that is narrow like L, but that includes or is inuenced by aset that is not constructible. This gives rise to the concept of relative constructibility, of which there are two avors,denoted L(A) and L[A].The class L(A) for a non-constructible set A is the intersection of all classes that are standard models of set theoryand contain A and all the ordinals.L(A) is dened by transnite recursion as follows:

    L0(A) = the smallest transitive set containing A as an element, i.e. the transitive closure of {A}. L(A) = Def (L(A)) If is a limit ordinal, then L(A) =

    S

  • 8 CHAPTER 2. CONSTRUCTIBLE UNIVERSE

    2.12 References Barwise, Jon (1975). Admissible Sets and Structures. Berlin: Springer-Verlag. ISBN 0-387-07451-1. Devlin, Keith J. (1984). Constructibility. Berlin: Springer-Verlag. ISBN 0-387-13258-9. Felgner, Ulrich (1971). Models of ZF-Set Theory. Lecture Notes in Mathematics. Springer-Verlag. ISBN3-540-05591-6.

    Gdel, Kurt (1938). TheConsistency of theAxiom ofChoice and of theGeneralizedContinuum-Hypothesis.Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sci-ences) 24 (12): 556557. doi:10.1073/pnas.24.12.556. JSTOR 87239. PMC 1077160. PMID 16577857.

    Gdel, Kurt (1940). The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies 3. Prince-ton, N. J.: Princeton University Press. ISBN 978-0-691-07927-1. MR 0002514.

    Jech, Thomas (2002). Set Theory. Springer Monographs in Mathematics (3rd millennium ed.). Springer.ISBN 3-540-44085-2.

  • Chapter 3

    Grothendieck universe

    In mathematics, a Grothendieck universe is a set U with the following properties:

    1. If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.)2. If x and y are both elements of U, then {x,y} is an element of U.3. If x is an element of U, then P(x), the power set of x, is also an element of U.4. If fxg2I is a family of elements of U, and if I is an element of U, then the union

    S2I x is an element of

    U.

    AGrothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountableGrothendieck universes provide models of set theory with the natural -relation, natural powerset operation etc.)Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to AlexanderGrothendieck, who used them as a way of avoiding proper classes in algebraic geometry.The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of ZermeloFraenkel set theory;in particular it would imply the existence of strongly inaccessible cardinals. TarskiGrothendieck set theory is anaxiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a Grothendieckuniverse. The concept of a Grothendieck universe can also be dened in a topos. [1]

    3.1 PropertiesAs an example, we will prove an easy proposition.

    Proposition. If x 2 U and y x , then y 2 U .Proof. y 2 P (x) because y x . P (x) 2 U because x 2 U , so y 2 U .

    The axioms of Grothendieck universes imply that every set is an element of some Grothendieck universe.It is similarly easy to prove that any Grothendieck universe U contains:

    All singletons of each of its elements, All products of all families of elements of U indexed by an element of U, All disjoint unions of all families of elements of U indexed by an element of U, All intersections of all families of elements of U indexed by an element of U, All functions between any two elements of U, and All subsets of U whose cardinal is an element of U.

    In particular, it follows from the last axiom that if U is non-empty, it must contain all of its nite subsets and a subsetof each nite cardinality. One can also prove immediately from the denitions that the intersection of any class ofuniverses is a universe.

    9

  • 10 CHAPTER 3. GROTHENDIECK UNIVERSE

    3.2 Grothendieck universes and inaccessible cardinalsThere are two simple examples of Grothendieck universes:

    The empty set, and The set of all hereditarily nite sets V! .

    Other examples are more dicult to construct. Loosely speaking, this is because Grothendieck universes are equiv-alent to strongly inaccessible cardinals. More formally, the following two axioms are equivalent:

    (U) For each set x, there exists a Grothendieck universe U such that x U.(C) For each cardinal , there is a strongly inaccessible cardinal that is strictly larger than .

    To prove this fact, we introduce the function c(U). Dene:

    c(U) = supx2U

    jxj

    where by |x| we mean the cardinality of x. Then for any universe U, c(U) is either zero or strongly inaccessible.Assuming it is non-zero, it is a strong limit cardinal because the power set of any element of U is an element of Uand every element of U is a subset of U. To see that it is regular, suppose that c is a collection of cardinals indexedby I, where the cardinality of I and of each c is less than c(U). Then, by the denition of c(U), I and each c canbe replaced by an element of U. The union of elements of U indexed by an element of U is an element of U, so thesum of the c has the cardinality of an element of U, hence is less than c(U). By invoking the axiom of foundation,that no set is contained in itself, it can be shown that c(U) equals |U |; when the axiom of foundation is not assumed,there are counterexamples (we may take for example U to be the set of all nite sets of nite sets etc. of the sets xwhere the index is any real number, and x = {x} for each . Then U has the cardinality of the continuum, butall of its members have nite cardinality and so c(U) = @0 ; see Bourbakis article for more details).Let be a strongly inaccessible cardinal. Say that a set S is strictly of type if for any sequence sn ... s0 S, |sn|< . (S itself corresponds to the empty sequence.) Then the set u() of all sets strictly of type is a Grothendieckuniverse of cardinality . The proof of this fact is long, so for details, we again refer to Bourbakis article, listed inthe references.To show that the large cardinal axiom (C) implies the universe axiom (U), choose a set x. Let x0 = x, and for each n,let xn =S xn be the union of the elements of xn. Let y =Sn xn. By (C), there is a strongly inaccessible cardinal such that |y| < . Let u() be the universe of the previous paragraph. x is strictly of type , so x u(). To show thatthe universe axiom (U) implies the large cardinal axiom (C), choose a cardinal . is a set, so it is an element of aGrothendieck universe U. The cardinality of U is strongly inaccessible and strictly larger than that of .In fact, any Grothendieck universe is of the form u() for some . This gives another form of the equivalence betweenGrothendieck universes and strongly inaccessible cardinals:

    For any Grothendieck universe U, |U | is either zero, @0 , or a strongly inaccessible cardinal. And if iszero, @0 , or a strongly inaccessible cardinal, then there is a Grothendieck universe u(). Furthermore,u(|U |) = U, and |u()| = .

    Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of ZermeloFraenkel settheory (ZFC), the existence of universes other than the empty set andV! cannot be proved fromZFC either. However,strongly inaccessible cardinals are on the lower end of the list of large cardinals; thus, most set theories that use largecardinals (such as ZFC plus there is a measurable cardinal", ZFC plus there are innitely many Woodin cardinals")will prove that Grothendieck universes exist.

    3.3 See also Constructible universe Universe (mathematics) Von Neumann universe

  • 3.4. REFERENCES 11

    3.4 References[1] Streicher, Thomas (2006). Universes in Toposes (PDF). From Sets and Types to Topology and Analysis: Towards Prac-

    ticable Foundations for Constructive Mathematics. Clarendon Press. pp. 7890. ISBN 9780198566519.

    Bourbaki, Nicolas (1972). Univers. InMichael Artin, Alexandre Grothendieck, Jean-Louis Verdier, eds. Sminairede Gomtrie Algbrique du Bois Marie 196364 Thorie des topos et cohomologie tale des schmas (SGA 4) vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185217.

  • Chapter 4

    Gdel operation

    In mathematical set theory, a set of Gdel operations is a nite collection of operations on sets that can be usedto construct the constructible sets from ordinals. Gdel (1940) introduced the original set of 8 Gdel operations1,...,8 under the name fundamental operations. Other authors sometimes use a slightly dierent set of about 8to 10 operations, usually denoted G1, G2,...

    4.1 DenitionGdel (1940) used the following eight operations as a set of Gdel operations (which he called fundamental opera-tions):

    1. F1(X;Y ) = fX;Y g2. F2(X;Y ) = E X = f(a; b) 2 X j a 2 bg3. F3(X;Y ) = X Y4. F4(X;Y ) = X Y = X (V Y ) = f(a; b) 2 X j b 2 Y g5. F5(X;Y ) = X D(Y ) = fb 2 X j 9a(a; b) 2 Y g6. F6(X;Y ) = X Y 1 = f(a; b) 2 X j (b; a) 2 Y g7. F7(X;Y ) = X Cnv2(Y ) = f(a; b; c) 2 X j (a; c; b) 2 Y g8. F8(X;Y ) = X Cnv3(Y ) = f(a; b; c) 2 X j (c; a; b) 2 Y g

    The second expression in each line gives Gdels denition in his original notation, where the dot means intersection,V is the universe, E is the membership relation, and so on.Jech (2003) uses the following set of 10 Gdel operations.

    1. G1(X;Y ) = fX;Y g2. G2(X;Y ) = X Y3. G3(X;Y ) = f(x; y) j x 2 X; y 2 Y; x 2 yg4. G4(X;Y ) = X Y5. G5(X;Y ) = X \ Y6. G6(X) = [X7. G7(X) = dom(X)8. G8(X) = f(x; y) j (y; x) 2 Xg9. G9(X) = f(x; y; z) j (x; z; y) 2 Xg10. G10(X) = f(x; y; z) j (y; z; x) 2 Xg

    12

  • 4.2. PROPERTIES 13

    4.2 PropertiesGdels normal form theorem states that if (x1,...xn) is a formula with all quantiers bounded, then the function{(x1,...,xn) X1...Xn | (x1, ..., xn)) of X1, ..., Xn is given by a composition of some Gdel operations.

    4.3 References Gdel, Kurt (1940). The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies 3. Prince-ton, N. J.: Princeton University Press. ISBN 978-0-691-07927-1. MR 0002514.

    Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

  • Chapter 5

    Jensen hierarchy

    In set theory, amathematical discipline, the Jensen hierarchy or J-hierarchy is amodication ofGdel's constructiblehierarchy, L, that circumvents certain technical diculties that exist in the constructible hierarchy. The J-Hierarchygures prominently in ne structure theory, a eld pioneered by Ronald Jensen, for whom the Jensen hierarchy isnamed.

    5.1 DenitionAs in the denition of L, let Def(X) be the collection of sets denable with parameters over X:

    Def(X) = { {y | y X and (y, z1, ..., zn) is true in (X, )} | is a rst order formula and z1, ..., zn areelements of X}.

    The constructible hierarchy, L is dened by transnite recursion. In particular, at successor ordinals, L = Def(L).The diculty with this construction is that each of the levels is not closed under the formation of unordered pairs; fora given x, y L L, the set {x,y} will not be an element of L, since it is not a subset of L.However, L does have the desirable property of being closed under 0 separation.Jensens modied hierarchy retains this property and the slightly weaker condition that J+1 \ Pow(J) = Def(J), but is also closed under pairing. The key technique is to encode hereditarily denable sets over J by codes; thenJ will contain all sets whose codes are in J.Like L, J is dened recursively. For each ordinal , we dene Wn to be a universal predicate for J. Weencode hereditarily denable sets asX(n+ 1; e) = fX(n; f) jWn+1(e; f)g , withX(0; e) = e . Then set J, to be {X(n, e) | e in J}. Finally, J = Sn2! J;n .5.2 PropertiesEach sublevel J, n is transitive and contains all ordinals less than or equal to + n. The sequence of sublevels isstrictly increasing in n, since a m predicate is also n for any n > m. The levels J will thus be transitive and strictlyincreasing as well, and are also closed under pairing, Delta-0 comprehension and transitive closure. Moreover, theyhave the property that

    J+1 \ Pow(J) = Def(J);as desired.The levels and sublevels are themselves 1 uniformly denable [i.e. the denition of J, n in J does not dependon ], and have a uniform 1 well-ordering. Finally, the levels of the Jensen hierarchy satisfy a condensation lemmamuch like the levels of Godels original hierarchy.

    14

  • 5.3. RUDIMENTARY FUNCTIONS 15

    5.3 Rudimentary functionsA rudimentary function is a function that can be obtained from the following operations:

    F(x1, x2, ...) = xi is rudimentary F(x1, x2, ...) = {xi, xj} is rudimentary F(x1, x2, ...) = xi xj is rudimentary Any composition of rudimentary functions is rudimentary zyG(z, x1, x2, ...) is rudimentary

    For any set M let rud(M) be the smallest set containing M{M} closed under the rudimentary operations. Then theJensen hierarchy satises J = rud(J).

    5.4 References Sy Friedman (2000) Fine Structure and Class Forcing, Walter de Gruyter, ISBN 3-11-016777-8

  • Chapter 6

    Minimal model (set theory)

    In set theory, a minimal model is a minimal standard model of ZFC. Minimal models were introduced by (Shep-herdson 1951, 1952, 1953).The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows fromthe existence of a standard model as follows. If there is a set W in the von Neumann universe V which is a standardmodel of ZF, and the ordinal is the set of ordinals which occur in W, then L is the class of constructible sets ofW. If there is a set which is a standard model of ZF, then the smallest such set is such a L. This set is called theminimal model of ZFC, and also satises the axiom of constructibility V=L. The downward LwenheimSkolemtheorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s ofthe minimal model can be named; in other words there is a rst order sentence (x) such that s is the unique elementof the minimal model for which (s) is true.Cohen (1963) gave another construction of the minimal model as the strongly constructible sets, using a modiedform of Godels constructible universe.Of course, any consistent theory must have a model, so even within the minimal model of set theory there are setswhich are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, theydo not use the normal element relation and they are not well founded.If there is no standard model then the minimal model cannot exist as a set. However in this case the class of allconstructible sets plays the same role as the minimal model and has similar properties (though it is now a proper classrather than a countable set).

    6.1 References Cohen, Paul J. (1963), Aminimalmodel for set theory, Bull. Amer. Math. Soc. 69: 537540, doi:10.1090/S0002-9904-1963-10989-1, MR 0150036

    Shepherdson, J. C. (1951), Inner models for set theory. I, The Journal of Symbolic Logic (Association forSymbolic Logic) 16 (3): 161190, doi:10.2307/2266389, JSTOR 2266389, MR 0045073

    Shepherdson, J. C. (1952), Inner models for set theory. II, The Journal of Symbolic Logic (Association forSymbolic Logic) 17 (4): 225237, doi:10.2307/2266609, JSTOR 2266609, MR 0053885

    Shepherdson, J. C. (1953), Inner models for set theory. III, The Journal of Symbolic Logic (Association forSymbolic Logic) 18 (2): 145167, doi:10.2307/2268947, JSTOR 2268947, MR 0057828

    16

  • Chapter 7

    Silver machine

    This article is about the kind of mathematical object. For the Hawkwind song, see Silver Machine. For the Vaporssong, see Silver Machines.

    In set theory, Silver machines are devices used for bypassing the use of ne structure in proofs of statements holdingin L. They were invented by set theorist Jack Silver as a means of proving global square holds in the constructibleuniverse.

    7.1 PreliminariesAn ordinal is *denable from a class of ordinals X if and only if there is a formula (0; 1; : : : ; n) and91; : : : ; n; 2 X such that is the unique ordinal for which j=L (; 1 ; : : : ; n) where for all we de-ne to be the name for within L .A structure hX;1. X On .2. < is the ordering on On restricted to X.

    3. 8i; hi is a partial function from Xk(i) to X, for some integer k(i).

    If N = hX;= h2i (x1; : : : ; xk(i))

    7.2 Silver machineA Silver machine is an eligible structure of the formM = hOn;M+1[A] M[(A \ ) [H] [ fg

    Skolem property. If is *denable from the set X On , then 2 M [X] ; moreover there is an ordinal