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 Lα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 reflection principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 The generalized continuum hypothesis holds in L . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 Constructible sets are definable 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 Gödel operation 124.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Jensen hierarchy 145.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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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 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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, first investigated by Kurt Gödel, 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 Implications

The axiom of constructibility implies the axiom of choice over Zermelo–Fraenkel set theory. It also settles manynatural mathematical questions independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Forexample, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin’s hypoth-esis, and the existence of an analytical (in fact,∆1

2 ) 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 ω1-Erdős 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 sufficiently 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

“Gödel universe” redirects here. For Kurt Gödel’s cosmological solution to the Einstein field equations, see Gödelmetric.

In mathematics, in set theory, the constructible universe (or Gödel’s constructible universe), denoted L, is aparticular class of sets that can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel 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 Neumann’s universe, at a successor stage, one takes Vα₊₁ to be the set of all subsets of the previousstage, Vα. By contrast, in Gödel’s constructible universe L, one uses only those subsets of the previous stage that are:

• definable by a formula in the formal language of set theory

• with parameters from the previous stage and

• with the quantifiers interpreted to range over the previous stage.

By limiting oneself to sets defined 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.Define

Def(X) :={{y | y ∈ X and (X,∈) |= Φ(y, z1, . . . , zn)}

∣∣∣ Φ and formula first-order a is z1, . . . , zn ∈ X}.

L is defined by transfinite recursion as follows:

• L0 := ∅.

• Lα+1 := Def(Lα).

• If λ is a limit ordinal, then Lλ :=∪

α<λ Lα. Here α<λ means α precedes λ.

• L :=∪

α∈Ord Lα. Here Ord denotes the class of all ordinals.

2

2.2. ADDITIONAL FACTS ABOUT THE SETS LΑ 3

If z is an element of Lα, then z = {y | y ∈ Lα and y ∈ z} ∈ Def (Lα) = Lα₊₁. So Lα is a subset of Lα₊₁, which is asubset of the power set of Lα. 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 Lα

An equivalent definition for Lα is:

Lα =∪β<α

Def(Lβ)

For any finite ordinal n, the sets L and V are the same (whether V equals L or not), and thus Lω = Vω: theirelements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC inwhich V equals L, Lω₊₁ is a proper subset of Vω₊₁, and thereafter Lα₊₁ is a proper subset of the power set of Lα forall α > ω. On the other hand, V equals L does imply that Vα equals Lα if α = ωα, for example if α is inaccessible.More generally, V equals L implies Hα equals Lα for all infinite cardinals α.If α is an infinite ordinal then there is a bijection between Lα and α, and the bijection is constructible. So these setsare equinumerous in any model of set theory that includes them.As defined above, Def(X) is the set of subsets of X defined by Δ0 formulas (that is, formulas of set theory containingonly bounded quantifiers) that use as parameters only X and its elements.An alternate definition, due to Gödel, characterizes each Lα₊₁ as the intersection of the power set of Lα with theclosure of Lα∪{Lα} under a collection of nine explicit functions. This definition makes no reference to definability.All arithmetical subsets of ω and relations on ω belong to Lω₊₁ (because the arithmetic definition gives one in Lω₊₁).Conversely, any subset of ω belonging to Lω₊₁ is arithmetical (because elements of Lω can be coded by naturalnumbers in such a way that ∈ is definable, i.e., arithmetic). On the other hand, Lω₊₂ already contains certain non-arithmetical subsets of ω, such as the set of (natural numbers coding) true arithmetical statements (this can be definedfrom Lω₊₁ so it is in Lω₊₂).All hyperarithmetical subsets of ω and relations on ω belong to LωCK

1(where ωCK

1 stands for the Church-Kleeneordinal), and conversely any subset of ω that belongs to LωCK

1is hyperarithmetical.[2]

2.3 L is a standard inner model of ZFC

L 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 satisfies 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 x∈L, thenby the transitivity of L, y∈L. 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 L’s 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 | y∈L0 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 x∈L and y∈L, then there is some ordinal α such that x∈Lα and y∈Lα. Then {x,y} = {s | s∈Lα and(s=x or s=y)} ∈ Lα₊₁. 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 ∈ Lα, then its elements are in Lα and their elements are also in Lα. So y is a subset of Lα. y = {s |s∈Lα and there exists z∈x such that s∈z} ∈ Lα₊₁. Thus y ∈ L.

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

From transfinite induction, we get that each ordinal α ∈ Lα₊₁. In particular, ω ∈ Lω₊₁ and thus ω ∈ L.

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

By induction on subformulas of P, one can show that there is an α such that Lα contains S and z1,...,zand (P is true in Lα if and only if P is true in L (this is called the "reflection principle")). So {x | x∈S andP(x,z1,...,z ) holds in L} = {x | x∈Lα and x∈S and P(x,z1,...,z ) holds in Lα} ∈ Lα₊₁. Thus the subset isin L.

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

Let Q(x,y) be the formula that relativizes P to L, i.e. all quantifiers in P are restricted to L. Q is a muchmore complex formula than P, but it is still a finite 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 | y∈L and there exists x∈S suchthat P(x,y) holds in L} = {y | there exists x∈S 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 α such that this set is a subsetof Lα ∈ Lα₊₁. Then one can use the axiom of separation in L to finish 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 α such that the intersection is a subset of Lα. Then theintersection is {z | z∈Lα and z is a subset of x} ∈ Lα₊₁. 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 definable well-ordering of L which definition 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 minimal

If W is any standard model of ZF sharing the same ordinals as V, then the L defined in W is the same as the L definedin V. In particular, Lα is the same in W and V, for any ordinal α. And the same formulas and parameters in Def (Lα)produce the same constructible sets in Lα₊₁.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 κ is the set of ordinals that occur in W,then Lκ 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 Lκ. Thisset is called the minimal model of ZFC. Using the downward Löwenheim–Skolem 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 Lκ and the V of Lκ are the real Lκ, we getthat V=L is true in L and in any Lκ that is a model of ZF. However, V=L does not hold in any other standard modelof ZF.

2.4.1 L and large cardinals

Since On⊂L⊆V, properties of ordinals that depend on the absence of a function or other structure (i.e. Π1ZF 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-ordered

There are various ways of well-ordering L. Some of these involve the “fine structure” of L, which was first describedby Ronald Bjorn Jensen in his 1972 paper entitled “The fine structure of the constructible hierarchy”. Instead ofexplaining the fine structure, we will give an outline of how L could be well-ordered using only the definition givenabove.Suppose x and y are two different sets in L and we wish to determine whether x<y or x>y. If x first appears in Lα₊₁and y first appears in Lᵦ₊₁ and β is different from α, then let x<y if and only if α<β. Henceforth, we suppose thatβ=α.Remember that Lα₊₁ = Def (Lα), which uses formulas with parameters from Lα to define the sets x and y. If onediscounts (for the moment) the parameters, the formulas can be given a standard Gödel numbering by the naturalnumbers. If Φ is the formula with the smallest Gödel number that can be used to define x, and Ψ is the formula withthe smallest Gödel number that can be used to define y, and Ψ is different from Φ, then let x<y if and only if Φ<Ψin the Gödel numbering. Henceforth, we suppose that Ψ=Φ.Suppose that Φ uses n parameters from Lα. Suppose z1,...,z is the sequence of parameters that can be used with Φto define x, and w1,...,w does the same for y. Then let x<y if and only if either z <w or (z =w and z -₁<w -₁) or(z =w and z -₁=w -₁ and z -₂<w -₂) or etc.. This is called the reverse-lexicographic ordering; if there are multiplesequences of parameters that define one of the sets, we choose the least one under this ordering. It being understood

6 CHAPTER 2. CONSTRUCTIBLE UNIVERSE

that each parameter’s possible values are ordered according to the restriction of the ordering of L to Lα, so thisdefinition involves transfinite recursion on α.The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite 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 Gödel numbers) of well-orderings. And L is well-ordered by the orderedsum (indexed by α) of the orderings on Lα₊₁.Notice that this well-ordering can be defined 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 reflection principle

Proving that the axiom of separation, axiom of replacement, and axiom of choice hold in L requires (at least as shownabove) the use of a reflection principle for L. Here we describe such a principle.By mathematical induction on n<ω, we can use ZF in V to prove that for any ordinal α, there is an ordinal β>α suchthat for any sentence P(z1,...,z ) with z1,...,z in Lᵦ and containing fewer than n symbols (counting a constant symbolfor an element of Lᵦ as one symbol) we get that P(z1,...,z ) holds in Lᵦ if and only if it holds in L.

2.7 The generalized continuum hypothesis holds in L

Let S ∈ Lα , and let T be any constructible subset of S. Then there is some β with T ∈ Lβ+1 , so T = {x ∈Lβ : x ∈ S ∧ Φ(x, zi)} = {x ∈ S : Φ(x, zi)} , for some formula Φ and some zi drawn from Lβ . By thedownward Löwenheim–Skolem theorem, there must be some transitive set K containing Lα and some wi , andhaving the same first-order theory as Lβ with the wi substituted for the zi ; and this K will have the same cardinalas Lα . Since V = L is true in Lβ , it is also true in K, so K = Lγ for some γ having the same cardinal as α. AndT = {x ∈ Lβ : x ∈ S ∧ Φ(x, zi)} = {x ∈ Lγ : x ∈ S ∧ Φ(x,wi)} because Lβ and Lγ have the same theory. SoT is in fact in Lγ+1 .So all the constructible subsets of an infinite set S have ranks with (at most) the same cardinal κ as the rank of S; itfollows that if α is the initial ordinal for κ+, then L∩P(S) ⊆ Lα+1 serves as the “powerset” of S within L. And thisin turn means that the “power set” of S has cardinal at most ||α||. Assuming S itself has cardinal κ, the “power set”must then have cardinal exactly κ+. But this is precisely the generalized continuum hypothesis relativized to L.

2.8 Constructible sets are definable from the ordinals

There is a formula of set theory that expresses the idea that X=Lα. It has only free variables for X and α. Using thiswe can expand the definition of each constructible set. If s∈Lα₊₁, then s = {y|y∈Lα and Φ(y,z1,...,z ) holds in (Lα,∈)}for some formula Φ and some z1,...,z in Lα. This is equivalent to saying that: for all y, y∈s if and only if [there existsX such that X=Lα and y∈X and Ψ(X,y,z1,...,z )] where Ψ(X,...) is the result of restricting each quantifier in Φ(...) toX. Notice that each z ∈Lᵦ₊₁ for some β<α. Combine formulas for the z’s with the formula for s and apply existentialquantifiers over the z’s outside and one gets a formula that defines the constructible set s using only the ordinals α thatappear in expressions like X=Lα as parameters.Example: The set {5,ω} is constructible. It is the unique set, s, that satisfies the formula:∀y(y ∈ s ⇐⇒ (y ∈ Lω+1 ∧ (∀a(a ∈ y ⇐⇒ a ∈ L5 ∧Ord(a)) ∨ ∀b(b ∈ y ⇐⇒ b ∈ Lω ∧Ord(b))))) ,where Ord(a) is short for:∀c ∈ a(∀d ∈ c(d ∈ a ∧ ∀e ∈ d(e ∈ c))).Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would

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 constructibility

Sometimes it is desirable to find a model of set theory that is narrow like L, but that includes or is influenced by aset that is not constructible. This gives rise to the concept of relative constructibility, of which there are two flavors,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 defined by transfinite recursion as follows:

• L0(A) = the smallest transitive set containing A as an element, i.e. the transitive closure of {A}.

• Lα₊₁(A) = Def (Lα(A))

• If λ is a limit ordinal, then Lλ(A) =∪

α<λ Lα(A) .

• L(A) =∪

α Lα(A) .

If L(A) contains a well-ordering of the transitive closure of {A}, then this can be extended to a well-ordering of L(A).Otherwise, the axiom of choice will fail in L(A).A common example is L(R), the smallest model that contains all the real numbers, which is used extensively inmoderndescriptive set theory.The class L[A] is the class of sets whose construction is influenced by A, where A may be a (presumably non-constructible) set or a proper class. The definition of this class uses DefA (X), which is the same as Def (X) exceptinstead of evaluating the truth of formulas Φ in the model (X,∈), one uses the model (X,∈,A) where A is a unarypredicate. The intended interpretation of A(y) is y∈A. Then the definition of L[A] is exactly that of L only with Defreplaced by DefA.L[A] is always a model of the axiom of choice. Even if A is a set, A is not necessarily itself a member of L[A],although it always is if A is a set of ordinals.It is essential to remember that the sets in L(A) or L[A] are usually not actually constructible and that the propertiesof these models may be quite different from the properties of L itself.

2.10 See also• Axiom of constructibility

• Statements true in L

• Reflection principle

• Axiomatic set theory

• Transitive set

• L(R)

• Ordinal definable

2.11 Notes[1] Gödel, 1938

[2] Barwise 1975, page 60 (comment following proof of theorem 5.9)

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.

• Gödel, 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): 556–557. doi:10.1073/pnas.24.12.556. JSTOR 87239. PMC 1077160. PMID 16577857.

• Gödel, 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 {xα}α∈I is a family of elements of U, and if I is an element of U, then the union

∪α∈I xα 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 Zermelo–Fraenkel set theory;in particular it would imply the existence of strongly inaccessible cardinals. Tarski–Grothendieck 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 defined in a topos. [1]

3.1 Properties

As an example, we will prove an easy proposition.

Proposition. If x ∈ U and y ⊆ x , then y ∈ U .Proof. y ∈ P (x) because y ⊆ x . P (x) ∈ U because x ∈ U , so y ∈ 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 finite subsets and a subsetof each finite cardinality. One can also prove immediately from the definitions that the intersection of any class ofuniverses is a universe.

9

10 CHAPTER 3. GROTHENDIECK UNIVERSE

3.2 Grothendieck universes and inaccessible cardinals

There are two simple examples of Grothendieck universes:

• The empty set, and• The set of all hereditarily finite sets Vω .

Other examples are more difficult 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 κ, there is a strongly inaccessible cardinal λ that is strictly larger than κ.

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

c(U) = supx∈U

|x|

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 cλ is a collection of cardinals indexedby I, where the cardinality of I and of each cλ is less than c(U). Then, by the definition of c(U), I and each cλ 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 cλ 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 finite sets of finite sets etc. of the sets xαwhere the index α is any real number, and xα = {xα} for each α. Then U has the cardinality of the continuum, butall of its members have finite cardinality and so c(U) = ℵ0 ; see Bourbaki’s article for more details).Let κ be a strongly inaccessible cardinal. Say that a set S is strictly of type κ if for any sequence sn ∈ ... ∈ s0 ∈ S, |sn|< κ. (S itself corresponds to the empty sequence.) Then the set u(κ) of all sets strictly of type κ is a Grothendieckuniverse of cardinality κ. The proof of this fact is long, so for details, we again refer to Bourbaki’s 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₊₁ =

∪xn be the union of the elements of xn. Let y =

∪n xn. By (C), there is a strongly inaccessible cardinal κ

such that |y| < κ. Let u(κ) be the universe of the previous paragraph. x is strictly of type κ, so x ∈ u(κ). To show thatthe universe axiom (U) implies the large cardinal axiom (C), choose a cardinal κ. κ 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 κ.In fact, any Grothendieck universe is of the form u(κ) for some κ. This gives another form of the equivalence betweenGrothendieck universes and strongly inaccessible cardinals:

For any Grothendieck universe U, |U | is either zero, ℵ0 , or a strongly inaccessible cardinal. And if κ iszero, ℵ0 , or a strongly inaccessible cardinal, then there is a Grothendieck universe u(κ). Furthermore,u(|U |) = U, and |u(κ)| = κ.

Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of Zermelo–Fraenkel settheory (ZFC), the existence of universes other than the empty set andVω 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 infinitely 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. 78–90. ISBN 9780198566519.

Bourbaki, Nicolas (1972). “Univers”. InMichael Artin, Alexandre Grothendieck, Jean-Louis Verdier, eds. Séminairede Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) –vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185–217.

Chapter 4

Gödel operation

In mathematical set theory, a set of Gödel operations is a finite collection of operations on sets that can be usedto construct the constructible sets from ordinals. Gödel (1940) introduced the original set of 8 Gödel operations𝔉1,...,𝔉8 under the name fundamental operations. Other authors sometimes use a slightly different set of about 8to 10 operations, usually denoted G1, G2,...

4.1 Definition

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

1. F1(X,Y ) = {X,Y }

2. F2(X,Y ) = E ·X = {(a, b) ∈ X | a ∈ b}

3. F3(X,Y ) = X − Y

4. F4(X,Y ) = X ↾ Y = X · (V × Y ) = {(a, b) ∈ X | b ∈ Y }

5. F5(X,Y ) = X ·D(Y ) = {b ∈ X | ∃a(a, b) ∈ Y }

6. F6(X,Y ) = X · Y −1 = {(a, b) ∈ X | (b, a) ∈ Y }

7. F7(X,Y ) = X · Cnv2(Y ) = {(a, b, c) ∈ X | (a, c, b) ∈ Y }

8. F8(X,Y ) = X · Cnv3(Y ) = {(a, b, c) ∈ X | (c, a, b) ∈ Y }

The second expression in each line gives Gödel’s definition 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 Gödel operations.

1. G1(X,Y ) = {X,Y }

2. G2(X,Y ) = X × Y

3. G3(X,Y ) = {(x, y) | x ∈ X, y ∈ Y, x ∈ y}

4. G4(X,Y ) = X − Y

5. G5(X,Y ) = X ∩ Y

6. G6(X) = ∪X

7. G7(X) = dom(X)

8. G8(X) = {(x, y) | (y, x) ∈ X}

9. G9(X) = {(x, y, z) | (x, z, y) ∈ X}

10. G10(X) = {(x, y, z) | (y, z, x) ∈ X}

12

4.2. PROPERTIES 13

4.2 Properties

Gödel’s normal form theorem states that if φ(x1,...xn) is a formula with all quantifiers bounded, then the function{(x1,...,xn) ∈ X1×...×Xn | φ(x1, ..., xn)) of X1, ..., Xn is given by a composition of some Gödel operations.

4.3 References• Gödel, 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 amodification ofGödel's constructiblehierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchyfigures prominently in fine structure theory, a field pioneered by Ronald Jensen, for whom the Jensen hierarchy isnamed.

5.1 Definition

As in the definition of L, let Def(X) be the collection of sets definable with parameters over X:

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

The constructible hierarchy, L is defined by transfinite recursion. In particular, at successor ordinals, Lα₊₁ = Def(Lα).The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; fora given x, y ε Lα₊₁ − Lα, the set {x,y} will not be an element of Lα₊₁, since it is not a subset of Lα.However, Lα does have the desirable property of being closed under Σ0 separation.Jensen’s modified hierarchy retains this property and the slightly weaker condition that Jα+1 ∩ Pow(Jα) = Def(Jα), but is also closed under pairing. The key technique is to encode hereditarily definable sets over Jα by codes; thenJα₊₁ will contain all sets whose codes are in Jα.Like Lα, Jα is defined recursively. For each ordinal α, we define Wα

n to be a universal Σ predicate for Jα. Weencode hereditarily definable sets asXα(n+ 1, e) = {X(n, f) | Wα

n+1(e, f)} , withXα(0, e) = e . Then set Jα,to be {X(n, e) | e in Jα}. Finally, Jα₊₁ =

∪n∈ω Jα,n .

5.2 Properties

Each sublevel Jα, n is transitive and contains all ordinals less than or equal to αω + n. The sequence of sublevels isstrictly increasing in n, since a Σm predicate is also Σn for any n > m. The levels Jα 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

Jα+1 ∩ Pow(Jα) = Def(Jα),

as desired.The levels and sublevels are themselves Σ1 uniformly definable [i.e. the definition of Jα, n in Jβ does not dependon β], and have a uniform Σ1 well-ordering. Finally, the levels of the Jensen hierarchy satisfy a condensation lemmamuch like the levels of Godel’s original hierarchy.

14

5.3. RUDIMENTARY FUNCTIONS 15

5.3 Rudimentary functions

A 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

• ∪z∈yG(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 satisfies Jα₊₁ = rud(Jα).

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 κ is the set of ordinals which occur in W, then Lκ 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 Lκ. This set is called theminimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolemtheorem 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 first order sentence φ(x) such that s is the unique elementof the minimal model for which φ(s) is true.Cohen (1963) gave another construction of the minimal model as the strongly constructible sets, using a modifiedform of Godel’s 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: 537–540, 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): 161–190, 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): 225–237, 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): 145–167, 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 fine 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 Preliminaries

An ordinal α is *definable from a class of ordinals X if and only if there is a formula ϕ(µ0, µ1, . . . , µn) and∃β1, . . . , βn, γ ∈ X such that α is the unique ordinal for which |=Lγ

ϕ(α◦, β◦1 , . . . , β

◦n) where for all α we de-

fine α◦ to be the name for α within Lγ .A structure ⟨X,<, (hi)i<ω⟩ is eligible if and only if:

1. X ⊆ On .

2. < is the ordering on On restricted to X.

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

If N = ⟨X,<, (hi)i<ω⟩ is an eligible structure then Nλ is defined to be as before but with all occurrences of Xreplaced withX ∩ λ .Let N1, N2 be two eligible structures which have the same function k. Then we say N1 ◁ N2 if ∀i ∈ ω and∀x1, . . . , xk(i) ∈ X1 we have:h1i (x1, . . . , xk(i)) ∼= h2

i (x1, . . . , xk(i))

7.2 Silver machine

A Silver machine is an eligible structure of the formM = ⟨On,<, (hi)i<ω⟩ which satisfies the following conditions:Condensation principle. If N ◁Mλ then there is an α such that N ∼= Mα .Finiteness principle. For each λ there is a finite setH ⊆ λ such that for any set A ⊆ λ+ 1 we have

Mλ+1[A] ⊆ Mλ[(A ∩ λ) ∪H] ∪ {λ}

Skolem property. If α is *definable from the set X ⊆ On , then α ∈ M [X] ; moreover there is an ordinal λ <[sup(X) ∪ α]+ , uniformly Σ1 definable from X ∪ {α} , such that α ∈ Mλ[X] .

17

18 CHAPTER 7. SILVER MACHINE

7.3 References• Keith J Devlin (1984). “Chapter IX”. Constructibility. ISBN 0-387-13258-9. - Please note that errors have beenfound in some results in this book concerning Kripke Platek set theory.

Chapter 8

Statements true in L

Here is a list of propositions that hold in the constructible universe (denoted L):

• The generalized continuum hypothesis and as a consequence

• The axiom of choice

• Diamondsuit

• Clubsuit

• Global square

• The existence of morasses

• The negation of the Suslin hypothesis

• The non-existence of 0# and as a consequence

• The non existence of all large cardinals which imply the existence of a measurable cardinal

• The truth of Whitehead’s conjecture that every abelian group A with Ext1(A, Z) = 0 is a free abelian group.

• The existence of a definable well-order of all sets (the formula for which can be given explicitly). In particular,L satisfies V=HOD.

Accepting the axiom of constructibility (which asserts that every set is constructible) these propositions also hold inthe von Neumann universe, resolving many propositions in set theory and some interesting questions in analysis.

19

Chapter 9

Von Neumann universe

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets,denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkelset theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of allmembers of the set.[1] In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself.The sets in V are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank.

9.1 Definition

The cumulative hierarchy is a collection of sets Vα indexed by the class of ordinal numbers, in particular, Vα is theset of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α; Vα may be defined bytransfinite recursion as follows:

• Let V0 be the empty set, {}:V0 := {}.

• For any ordinal number β, let Vᵦ₊₁ be the power set of Vᵦ:Vβ+1 := P(Vβ).

• For any limit ordinal λ, let Vλ be the union of all the V-stages so far:

Vλ :=∪β<λ

Vβ .

A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines “the setx is in Vα".The sets Vα are called stages or ranks.The class V is defined to be the union of all the V-stages:

V :=∪α

Vα.

An equivalent definition sets

Vα :=∪β<α

P(Vβ)

for each ordinal α, where P(X)is the powerset of X .The rank of a set S is the smallest α such that S ⊆ Vα .

20

9.2. APPLICATIONS AND INTERPRETATIONS 21

9.1.1 Finite and low cardinality stages of the hierarchy

The first five von Neumann stages V0 to V4 may be visualized as follows. (An empty box represents the empty set.A box containing only an empty box represents the set containing only the empty set, and so forth.)

First 5 von Neumann stages

The set V5 contains 216=65536 elements. The set V6 contains 265536 elements, which very substantially exceedsthe number of atoms in the known universe. So the finite stages of the cumulative hierarchy cannot be written downexplicitly after stage 5. The set Vω has the same cardinality as ω. The set Vω₊₁ has the same cardinality as the setof real numbers.

9.2 Applications and interpretations

9.2.1 Applications of V as models for set theories

If ω is the set of natural numbers, then Vω is the set of hereditarily finite sets, which is a model of set theory withoutthe axiom of infinity.[2][3]

Vω₊ω is the universe of “ordinary mathematics”, and is a model of Zermelo set theory.[4] A simple argument infavour of the adequacy of Vω₊ω is the observation that Vω₊₁ is adequate for the integers, while Vω₊₂ is adequatefor the real numbers, and most other normal mathematics can be built as relations of various kinds from these setswithout needing the axiom of replacement to go outside Vω₊ω.If κ is an inaccessible cardinal, then Vκ is a model of Zermelo-Fraenkel set theory (ZFC) itself, and Vκ₊₁ is a modelof Morse–Kelley set theory.[5][6] (Note that every ZFC model is also a ZF model, and every ZF model is also a Zmodel.)

9.2.2 Interpretation of V as the “set of all sets”

V is not “the set of all sets" for two reasons. First, it is not a set; although each individual stage Vα is a set, their unionV is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity)demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systemsmay omit the axiom of foundation or replace it by a strong negation (an example is Aczel’s anti-foundation axiom).These non-well-founded set theories are not commonly employed, but are still possible to study.A third objection to the “set of all sets” interpretation is that not all sets are necessarily “pure sets”, which are con-structed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of urelements,from which he constructed a transfinite recursive hierarchy in 1930.[7] Such urelements are used extensively in modeltheory, particularly in Fraenkel-Mostowski models.[8]

9.2.3 V and the axiom of regularity

The formulaV =⋃αVα is often considered to be a theorem, not a definition.[9][10] Roitman states (without references)that the realization that the axiom of regularity is equivalent to the equality of the universe of ZF sets to the cumulativehierarchy is due to von Neumann.[11]

22 CHAPTER 9. VON NEUMANN UNIVERSE

9.2.4 The existential status of V

Since the classV may be considered to be the arena for most of mathematics, it is important to establish that it “exists”in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistencyquestion, that is, whether the concept is free of contradictions. A major obstacle is posed by Gödel’s incompletenesstheorems, which effectively imply the impossibility of proving the consistency of ZF set theory.[12]

The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which actas the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbersand the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to restupon von Neumann’s 1923 and 1928 papers.[13] The integrity of the construction of V by transfinite induction maybe said to have then been established in Zermelo’s 1930 paper.[7]

9.3 History

The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982)to be inaccurately attributed to von Neumann.[14] The first publication of the von Neumann universe was by ErnstZermelo in 1930.[7]

Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by vonNeumann for both Zermelo-Fraenkel set theory[15] and Neumann’s own set theory (which later developed into NBGset theory).[16] In neither of these papers did he apply his transfinite recursive method to construct the universe ofall sets. The presentations of the von Neumann universe by Bernays[9] and Mendelson[10] both give credit to vonNeumann for the transfinite induction construction method, although not for its application to the construction of theuniverse of ordinary sets.The notation V is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano,the letter V signifying “Verum”, which he used both as a logical symbol and to denote the class of all individuals.[17]Peano’s notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.[18] The V notation(for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfiniteinduction. Paul Cohen[19] explicitly attributes his use of the letter V (for the class of all sets) to a 1940 paper byGödel,[20] although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.[18]

9.4 See also• Universe (mathematics)

• Constructible universe

• Grothendieck universe

• Inaccessible cardinal

• S (set theory)

• John von Neumann

9.5 Notes[1] Mirimanoff 1917; Moore 2013, pp. 261-262; Rubin 1967, p. 214.

[2] Roitman 2011, p. 136, proves that: "Vω is a model of all of the axioms of ZFC except infinity.”

[3] Cohen 2008, p. 54, states: “The first really interesting axiom [of ZF set theory] is the Axiom of Infinity. If we drop it,then we can take as a model for ZF the set M of all finite sets which can be built up from ∅. [...] It is clear that M will bea model for the other axioms, since none of these lead out of the class of finite sets.”

[4] Smullyan & Fitting 2010. See page 96 for proof that Vω₊ω is a Zermelo model.

[5] Cohen 2008, p. 80, states and justifies that if κ is strongly inaccessible, then Vκ is a model of ZF.

9.6. REFERENCES 23

“It is clear that if A is an inaccessible cardinal, then the set of all sets of rank less than A is a model for ZF,since the only two troublesome axioms, Power Set and Replacement, do not lead out of the set of cardinalsless than A.”

[6] Roitman 2011, pp. 134–135, proves that if κ is strongly inaccessible, then Vκ is a model of ZFC.

[7] Zermelo 1930. See particularly pages 36–40.

[8] Howard & Rubin 1998, pp. 175–221.

[9] Bernays 1991. See pages 203–209.

[10] Mendelson 1964. See page 202.

[11] Roitman 2011. See page 79.

[12] See article On Formally Undecidable Propositions of Principia Mathematica and Related Systems and Gödel 1931.

[13] von Neumann 1923, von Neumann 1928b. See also the English-language presentation of von Neumann’s “general recursiontheorem” by Bernays 1991, pp. 100–109.

[14] Moore 2013. See page 279 for the assertion of the false attribution to von Neumann. See pages 270 and 281 for theattribution to Zermelo.

[15] von Neumann 1928b.

[16] von Neumann 1928a. See pages 745–752.

[17] Peano 1889. See pages VIII and XI.

[18] Whitehead & Russell 2009. See page 229.

[19] Cohen 2008. See page 88.

[20] Gödel 1940.

9.6 References

• Bernays, Paul (1991) [1958]. Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.

• Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: DoverPublications. ISBN 978-0-486-46921-8.

• Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme,I”. Monatshefte für Mathematik und Physik 38: 173–198.

• Gödel, Kurt (1940). The consistency of the axiom of choice and of the generalized continuum-hypothesis withthe axioms of set theory. Annals of Mathematics Studies 3. Princeton, N. J.: Princeton University Press.

• Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: Amer-ican Mathematical Society. pp. 175–221. ISBN 9780821809778.

• Jech, Thomas (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN3-540-44085-2.

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

• Mendelson, Elliott (1964). Introduction to Mathematical Logic. Van Nostrand Reinhold.

• Mirimanoff, Dmitry (1917). “Les antinomies de Russell et de Burali-Forti et le probleme fondamental de latheorie des ensembles”. L'Enseignement Mathématique 19: 37–52.

• Moore, Gregory H (2013) [1982]. Zermelo’s axiom of choice: Its origins, development & influence. DoverPublications. ISBN 978-0-486-48841-7.

• Peano, Giuseppe (1889). Arithmetices principia: nova methodo exposita. External link in |title= (help)

24 CHAPTER 9. VON NEUMANN UNIVERSE

• Roitman, Judith (2011) [1990]. Introduction to Modern Set Theory. Virginia Commonwealth University. ISBN978-0-9824062-4-3.

• Rubin, Jean E. (1967). Set Theory for the Mathematician. San Francisco: Holden-Day. ASIN B0006BQH7S.

• Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originallypublished in 1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover.ISBN 978-0-486-47484-7.

• von Neumann, John (1923). “Zur Einführung der transfiniten Zahlen”. Acta litt. Acad. Sc. Szeged X. 1: 199–208.. English translation: van Heijenoort, Jean (1967), “On the introduction of transfinite numbers”, FromFrege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 346–354

• von Neumann, John (1928a). “Die Axiomatisierung der Mengenlehre”. Mathematische Zeitschrift 27: 669–752. doi:10.1007/bf01171122.

• von Neumann, John (1928b). "Über die Definition durch transfinite Induktion und verwandte Fragen derallgemeinen Mengenlehre”. Mathematische Annalen 99: 373–391. doi:10.1007/bf01459102.

• Whitehead, Alfred North; Russell, Bertrand (2009) [1910]. Principia Mathematica. Volume One. MerchantBooks. ISBN 978-1-60386-182-3.

• Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagender Mengenlehre”. Fundamenta Mathematicae 16: 29–47.

Chapter 10

Zero sharp

In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscerniblesand order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (usingGödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC,the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced asa set of formulae in Silver’s 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscoveredby Solovay (1967, p.52), who considered it as a subset of the natural numbers and introduced the notation O# (witha capital letter O; this later changed to a number 0).Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets,while if it does not exist then the universe of all sets is closely approximated by the constructible sets.

10.1 Definition

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constantsymbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentencesabout the constructible universe, with ci interpreted as the uncountable cardinal ℵi. (Here ℵi means ℵi in the fulluniverse, not the constructible universe.)There is a subtlety about this definition: by Tarski’s undefinability theorem it is not in general possible to define thetruth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existenceof a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible todefine the truth of statements about the constructible universe. More generally, the definition of 0# works providedthat there is an uncountable set of indiscernibles for some Lα, and the phrase “0# exists” is used as a shorthand wayof saying this.There are several minor variations of the definition of 0#, which make no significant difference to its properties. Thereare many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subsetof the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of thehereditarily finite sets, or as a real number.

10.2 Statements that imply the existence of 0#

The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existenceof ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0#implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot beused to prove the existence of 0#.Chang’s conjecture implies the existence of 0#.

25

26 CHAPTER 10. ZERO SHARP

10.3 Statements equivalent to existence of 0#

Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructibleuniverse L into itself.Donald A.Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightfaceanalytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#.It follows from Jensen’s covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in theconstructible universe L.Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent tothe existence of 0#.

10.4 Consequences of existence and non-existence

Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L andsatisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existenceof 0# contradicts the axiom of constructibility: V = L.If 0# exists, then it is an example of a non-constructible Δ13 set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ12 and Π12 sets of integers are constructible.On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonicalinner model that approximates the large cardinal structure of the universe considered. In that case, Jensen’s coveringlemma holds:

For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the samecardinality as x.

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannotbe removed. For example, consider Namba forcing, that preserves ω1 and collapses ω2 to an ordinal of cofinality ω. Let G be an ω -sequence cofinal on ωL

2 and generic over L. Then no set in L of L-size smaller than ωL2 (which is

uncountable in V, since ω1 is preserved) can cover G , since ω2 is a regular cardinal.

10.5 Other sharps

If x is any set, then x# is defined analogously to 0# except that one uses L[x] instead of L. See the section on relativeconstructibility in constructible universe.

10.6 See also

• 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurablecardinal.

10.7 References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations ofMathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

• Harrington, Leo (1978), “Analytic determinacy and 0#", The Journal of Symbolic Logic 43 (4): 685–693,doi:10.2307/2273508, ISSN 0022-4812, MR 518675

10.7. REFERENCES 27

• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

• Martin, Donald A. (1970), “Measurable cardinals and analytic games”, Polska Akademia Nauk. FundamentaMathematicae 66: 287–291, ISSN 0016-2736, MR 0258637

• Silver, Jack H. (1971) [1966], “Some applications of model theory in set theory”, Annals of Pure and AppliedLogic 3 (1): 45–110, doi:10.1016/0003-4843(71)90010-6, ISSN 0168-0072, MR 0409188

• Solovay, Robert M. (1967), “A nonconstructible Δ13 set of integers”, Transactions of the American Mathematical Society 127: 50–75, ISSN 0002-9947, JSTOR1994631, MR 0211873

28 CHAPTER 10. ZERO SHARP

10.8 Text and image sources, contributors, and licenses

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