Inner model theoryFrom Wikipedia, the free encyclopedia

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

2

Chapter 2

Chang’s model

In mathematical set theory, Chang’s model is the smallest inner model of set theory closed under countable se-quences. It was introduced by Chang (1971). More generally Chang introduced the smallest inner model closedunder taking sequences of length less than κ for any infinite cardinal κ. For κ countable this is the constructibleuniverse, and for κ the first uncountable cardinal it is Chang’s model.

2.1 References• Chang, C. C. (1971), “Sets constructible using Lκκ", Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII,Part I, Providence, R.I.: Amer. Math. Soc., pp. 1–8, MR 0280357, Zbl 0218.02061

3

Chapter 3

Code (set theory)

In set theory, a code for a hereditarily countable set

x ∈ Hℵ1

is a set

E ⊂ ω × ω

such that there is an isomorphism between (ω,E) and (X, ∈ ) where X is the transitive closure of {x}. If X is finite(with cardinality n), then use n×n instead of ω×ω and (n,E) instead of (ω,E).According to the axiom of extensionality, the identity of a set is determined by its elements. And since those elementsare also sets, their identities are determined by their elements, etc.. So if one knows the element relation restrictedto X, then one knows what x is. (We use the transitive closure of {x} rather than of x itself to avoid confusing theelements of x with elements of its elements or whatever.) A code includes that information identifying x and alsoinformation about the particular injection from X into ω which was used to create E. The extra information about theinjection is non-essential, so there are many codes for the same set which are equally useful.So codes are a way of mapping Hℵ1 into the powerset of ω×ω. Using a pairing function on ω (such as (n,k) goes to(n2+2·n·k+k2+n+3·k)/2), we can map the powerset of ω×ω into the powerset of ω. And we can map the powerset ofω into the Cantor set, a subset of the real numbers. So statements aboutHℵ1 can be converted into statements aboutthe reals. Consequently,Hℵ1 ⊂ L(R) .

Codes are useful in constructing mice.

3.1 See also• L(R)

3.2 References• William J.Mitchell,"The Complexity of the CoreModel”,"Journal of Symbolic Logic”,Vol.63,No.4,December1998,page 1393.

4

Chapter 4

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.

4.1 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.

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6 CHAPTER 4. CONSTRUCTIBLE UNIVERSE

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.

4.2 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]

4.3 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.

4.3. L IS A STANDARD INNER MODEL OF ZFC 7

{} = 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.

8 CHAPTER 4. CONSTRUCTIBLE UNIVERSE

4.4 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.

4.4.1 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.

4.5 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

4.6. L HAS A REFLECTION PRINCIPLE 9

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.

4.6 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.

4.7 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.

4.8 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

10 CHAPTER 4. CONSTRUCTIBLE UNIVERSE

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.

4.9 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.

4.10 See also• Axiom of constructibility

• Statements true in L

• Reflection principle

• Axiomatic set theory

• Transitive set

• L(R)

• Ordinal definable

4.11 Notes[1] Gödel, 1938

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

4.12. REFERENCES 11

4.12 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 5

Core model

In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to“the core model”, it is not a uniquely identified mathematical object. Rather, it is a class of inner models that underthe right set theoretic assumptions have very special properties, most notably covering properties. Intuitively, thecore model is “the largest canonical inner model there is” (Ernest Schimmerling and John R. Steel) and is typicallyassociated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase “core model below Φ" refers tothe definable inner model that exhibits the special properties under the assumption that there does not exist a cardinalsatisfying Φ. The core model program seeks to analyze large cardinal axioms by determining the core models belowthem.

5.1 History

The first core model was Kurt Gödel's constructible universe L. Ronald Jensen proved the covering lemma for L inthe 1970s under the assumption of the non-existence of zero sharp, establishing that L is the “core model below zerosharp”. The work of Solovay isolated another core model L[U], for U an ultrafilter on a measurable cardinal (and itsassociated “sharp”, zero dagger). Together with Tony Dodd, Jensen constructed the Dodd–Jensen core model (“thecore model below a measurable cardinal”) and proved the covering lemma for it and a generalized covering lemmafor L[U].Mitchell used coherent sequences of measures to develop core models containing multiple or higher-order measur-ables. Still later, the Steel core model used extenders and iteration trees to construct a core model below a Woodincardinal.

5.2 Construction of core models

Core models are constructed by transfinite recursion from small fragments of the core model called mice. An im-portant ingredient of the construction is the comparison lemma that allows giving a wellordering of the relevantmice.At the level of strong cardinals and above, one constructs an intermediate countably certified core model Kc, andthen, if possible, extracts K from Kc.

5.3 Properties of core models

K (and hence K) is a fine-structural countably iterable extender model below long extenders. (It is not currentlyknown how to deal with long extenders, which establish that a cardinal is superstrong.) Here countable iterabilitymeans ω1+1 iterability for all countable elementary substructures of initial segments, and it suffices to develop basictheory, including certain condensation properties. The theory of such models is canonical and well-understood. Theysatisfy GCH, the diamond principle for all stationary subsets of regular cardinals, the square principle (except atsubcompact cardinals), and other principles holding in L.

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5.4. CONSTRUCTION OF CORE MODELS 13

Kc is maximal in several senses. Kc computes the successors of measurable and many singular cardinals correctly.Also, it is expected that under an appropriate weakening of countable certifiability, Kc would correctly compute thesuccessors of all weakly compact and singular strong limit cardinals correctly. If V is closed under a mouse operator(an inner model operator), then so is Kc. Kc has no sharp: There is no natural non-trivial elementary embedding ofKc into itself. (However, unlike K, Kc may be elementarily self-embeddable.)If in addition there are also noWoodin cardinals in this model (except in certain specific cases, it is not known how thecore model should be defined if K has Woodin cardinals), we can extract the actual core model K. K is also its owncore model. K is locally definable and generically absolute: For every generic extension of V, for every cardinal κ>ω1

in V[G], K as constructed in H(κ) of V[G] equals K∩H(κ). (This would not be possible had K contained Woodincardinals). K is maximal, universal, and fully iterable. This implies that for every iterable extender model M (calleda mouse), there is an elementary embedding M→N and of an initial segment of K into N, and if M is universal, theembedding is of K into M.It is conjectured that if K exists and V is closed under a sharp operator M, then K is Σ1

1 correct allowing real numbersin K as parameters and M as a predicate. That amounts to Σ1

3 correctness (in the usual sense) if M is x→x#.The core model can also be defined above a particular set of ordinals X: X belongs to K(X), but K(X) satisfies theusual properties of K above X. If there is no iterable inner model with ω Woodin cardinals, then for some X, K(X)exists. The above discussion of K and Kc generalizes to K(X) and Kc(X).

5.4 Construction of core models

Conjecture:

• If there is no ω1+1 iterable model with long extenders (and hence models with superstrong cardinals), then Kc

exists.

• If Kc exists and as constructed in every generic extension of V (equivalently, under some generic collapseColl(ω, <κ) for a sufficiently large ordinal κ) satisfies “there are no Woodin cardinals”, then the Core Model Kexists.

Partial results for the conjecture are that:

1. If there is no inner model with a Woodin cardinal, then K exists.

2. If (boldface) Σ1 determinacy (n is finite) holds in every generic extension of V, but there is no iterable innermodel with n Woodin cardinals, then K exists.

3. If there is a measurable cardinal κ, then either Kc below κ exists, or there is an ω1+1 iterable model withmeasurable limit λ of both Woodin cardinals and cardinals strong up to λ.

If V has Woodin cardinals but not cardinals strong past a Woodin one, then under appropriate circumstances (acandidate for) K can be constructed by constructing K below eachWoodin cardinal (and below the class of all ordinals)κ but above that K as constructed below the supremum of Woodin cardinals below κ. The candidate core model isnot fully iterable (iterability fails at Woodin cardinals) or generically absolute, but otherwise behaves like K.

5.5 References• W.H. Woodin (2001). The Continuum Hypothesis, Part I. Notices of the AMS.

• William Mitchell. “Beginning Inner Model Theory” (being Chapter 17 in Volume 3 of “Handbook of SetTheory”) at .

• Matthew Foreman and Akihiro Kanamori (Editors). “Handbook of Set Theory”, Springer Verlag, 2010, ISBN978-1402048432.

• Ronald Jensen and John Steel. “K without the measurable”. Journal of Symbolic Logic Volume 78, Issue 3(2013), 708-734.

Chapter 6

Covering lemma

See also: Jensen’s covering theorem

In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinalsleads to the existence of a canonical inner model, called the core model, that is, in a sense, maximal and approximatesthe structure of the vonNeumann universeV. A covering lemma asserts that under some particular anti-large cardinalassumption, the core model exists and is maximal in a sense that depends on the chosen large cardinal.

6.1 Example

For example, if there is no inner model for a measurable cardinal, then the Dodd–Jensen core model, KDJ is the coremodel and satisfies the covering property, that is for every uncountable set x of ordinals, there is y such that y⊃x, yhas the same cardinality as x, and y ∈KDJ. (If 0# does not exist, then KDJ=L.)

6.2 Versions

If the core model K exists (and has no Woodin cardinals), then

1. If K has no ω1-Erdős cardinals, then for a particular countable (in K) and definable in K sequence of functionsfrom ordinals to ordinals, every set of ordinals closed under these functions is a union of a countable numberof sets in K. If L=K, these are simply the primitive recursive functions.

2. If K has no measurable cardinals, then for every uncountable set x of ordinals, there is y∈K such that x ⊂ y and|x|=|y|.

3. If K has only one measurable cardinal κ, then for every uncountable set x of ordinals, there is y∈K[C] suchthat x ⊂ y and |x|=|y|. Here C is either empty or Prikry generic over K (so it has order type ω and is cofinal inκ) and unique except up to a finite initial segment.

4. If K has no inaccessible limit of measurable cardinals and no proper class of measurable cardinals, then thereis a maximal and unique (except for a finite set of ordinals) set C (called a system of indiscernibles) for K suchthat for every sequence S in K of measure one sets consisting of one set for each measurable cardinal, C minus∪S is finite. Note that every κ\C is either finite or Prikry generic for K at κ except for members of C belowa measurable cardinal below κ. For every uncountable set x of ordinals, there is y∈K[C] such that x ⊂ y and|x|=|y|.

5. For every uncountable set x of ordinals, there is a set C of indiscernibles for total extenders on K such thatthere is y∈K[C] and x ⊂ y and |x|=|y|.

6. K computes the successors of singular and weakly compact cardinals correctly (Weak Covering Property).Moreover, if |κ|>ω1, then cofinality((κ+)K ) ≥ |κ|.

14

6.3. EXTENDERS AND INDESCERNIBLES 15

6.3 Extenders and indescernibles

For core models without overlapping total extenders, the systems of indescernibles are well-understood. Although(if K has an inaccessible limit of measurable cardinals), the system may depend on the set to be covered, it is well-determined and unique in a weaker sense. One application of the covering is counting the number of (sequencesof) indiscernibles, which gives optimal lower bounds for various failures of the singular cardinals hypothesis. Forexample, if K does not have overlapping total extenders, and κ is singular strong limit, and 2κ=κ++, then κ hasMitchell order at least κ++ in K. Conversely, a failure of the singular cardinal hypothesis can be obtained (in a genericextension) from κ with o(κ)=κ++.For core models with overlapping total extenders (that is with a cardinal strong up to a measurable one), the systemsof indiscernibles are poorly understood, and applications (such as the weak covering) tend to avoid rather than analyzethe indiscernibles.

6.4 Additional properties

If K exists, then every regular Jónsson cardinal is Ramsey in K. Every singular cardinal that is regular in K is mea-surable in K.Also, if the core model K(X) exists above a set X of ordinals, then it has the above discussed covering propertiesabove X.

6.5 References• Mitchell,William (2010), “The covering lemma”,Handbook of Set Theory, Springer, pp. 1497–1594, doi:10.1007/978-1-4020-5764-9_19, ISBN 978-1-4020-4843-2

Chapter 7

Extender (set theory)

In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing largecardinal properties. A nonprincipal ultrafilter is the most basic case of an extender.A (κ, λ)-extender can be defined as an elementary embedding of some modelM of ZFC− (ZFC minus the power setaxiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as acollection of ultrafilters, one for each n-tuple drawn from λ.

7.1 Formal definition of an extender

Let κ and λ be cardinals with κ≤λ. Then, a set E = {Ea|a ∈ [λ]<ω} is called a (κ,λ)-extender if the followingproperties are satisfied:

1. each Ea is a κ-complete nonprincipal ultrafilter on [κ]<ω and furthermore

(a) at least one Ea is not κ+-complete,(b) for each α ∈ κ , at least one Ea contains the set {s ∈ [κ]|a| : α ∈ s} .

2. (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).

3. (Normality) If f is such that {s ∈ [κ]|a| : f(s) ∈ max s} ∈ Ea , then for some b ⊇ a, {t ∈ κ|b| :(f ◦ πba)(t) ∈ t} ∈ Eb .

4. (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of theultrapowers Ult(V,Ea)).

By coherence, one means that if a and b are finite subsets of λ such that b is a superset of a, then if X is an elementof the ultrafilter Eb and one chooses the right way to project X down to a set of sequences of length |a|, then X isan element of Ea. More formally, for b = {α1, . . . , αn} , where α1 < · · · < αn < λ , and a = {αi1 , . . . , αim}, where m≤n and for j≤m the ij are pairwise distinct and at most n, we define the projection πba : {ξ1, . . . , ξn} 7→{ξi1 , . . . , ξim} (ξ1 < · · · < ξn) .Then Ea and Eb cohere if

X ∈ Ea ⇔ {s : πba(s) ∈ X} ∈ Eb

7.2 Defining an extender from an elementary embedding

Given an elementary embedding j:V→M, which maps the set-theoretic universe V into a transitive inner model M,with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines E = {Ea|a ∈ [λ]<ω} as follows:

16

7.3. REFERENCES 17

fora ∈ [λ]<ω, X ⊆ [κ]<ω : X ∈ Ea ⇔ a ∈ j(X).

One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.

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

• Jech, Thomas (2002). Set Theory (3rd ed.). Springer. ISBN 3-540-44085-2.

Chapter 8

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,...

8.1 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}

18

8.2. PROPERTIES 19

8.2 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.

8.3 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 9

Inner model

In mathematical logic, suppose T is a theory in the language

L = ⟨∈⟩

of set theory.If M is a model of L describing a set theory and N is a class of M such that

⟨N,∈M , . . .⟩

is a model of T containing all ordinals of M then we say that N is an inner model of T (in M).[1] Ordinarily thesemodels are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension ofV.This term inner model is sometimes applied to models that are proper classes; the term set model is used for modelsthat are sets.A model of set theory is called standard if the element relation of the model is the actual element relation restrictedto the model. A model is called transitive when it is standard and the base class is a transitive class of sets. A modelof set theory is often assumed to be transitive unless it is explicitly stated that it is non-standard. Inner models aretransitive, transitive models are standard, and standard models are well-founded.The assumption that there exists a standard model of ZFC (in a given universe) is stronger than the assumptionthat there exists a model. In fact, if there is a standard model, then there is a smallest standard model called theminimal model contained in all standard models. The minimal model contains no standard model (as it is minimal)but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem. Thismodel is necessarily not well founded otherwise its Mostowski collapse would be a standard model. (It is not wellfounded as a relation in the universe, though it satisfies the axiom of foundation so is “internally” well founded. Beingwell founded is not an absolute property.[2]) In particular in the minimal model there is a model of ZFC but there isno standard model of ZFC.

9.1 Use

Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension ofZFC (like ZFC + ∃ a measurable cardinal). When no theory is mentioned, it is usually assumed that the model underdiscussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of subtheories of ZFC(like ZF or KP) as well.

20

9.2. RELATED IDEAS 21

9.2 Related ideas

It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF (which is also an inner model ofZFC + GCH), called the constructible universe, or L.There is a branch of set theory called inner model theory that studies ways of constructing least inner models oftheories extending ZF. Inner model theory has led to the discovery of the exact consistency strength ofmany importantset theoretical properties.

9.3 References[1] Jech, Thomas (2002). Set Theory. Berlin: Springer-Verlag. ISBN 3-540-44085-2.

[2] Kunen, Kenneth (1980). Set Theory. Amsterdam: North-Holland Pub. Co. ISBN 0-444-86839-9., Page 117

9.4 See also• Countable transitive models and generic filters

Chapter 10

Inner model theory

In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof.Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of ageneric extension of V. Inner model theory studies the relationships of these models to determinacy, large cardinals,and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model theory.

10.1 Examples• The class of all sets is an inner model containing all other inner models.

• The first non-trivial example of an inner model was the constructible universe L developed by Kurt Gödel.Every model M of ZFC has an inner model LM satisfying the axiom of constructibility, and this will be thesmallest inner model ofM containing all the ordinals ofM. Regardless of the properties of the original model,LM will satisfy the generalized continuum hypothesis and combinatorial axioms such as the diamond principle◊.

• The sets that are hereditarily ordinal definable form an inner model

• The sets that are hereditarily definable over a countable sequence of ordinals form an inner model, used inSolovay’s theorem.

• L(R)

• L[U] (see zero dagger)

10.2 Consistency results

One important use of inner models is the proof of consistency results. If it can be shown that every model of anaxiom A has an inner model satisfying axiom B, then if A is consistent, B must also be consistent. This analysis ismost useful when A is an axiom independent of ZFC, for example a large cardinal axiom; it is one of the tools usedto rank axioms by consistency strength.

10.3 References• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag

• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-00384-7

22

Chapter 11

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.

11.1 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 .

11.2 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.

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24 CHAPTER 11. JENSEN HIERARCHY

11.3 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).

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

Chapter 12

L(R)

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals andall the reals.

12.1 Construction

It can be constructed in a manner analogous to the construction of L (that is, Gödel’s constructible universe), byadding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.

12.2 Assumptions

In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannotshow even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiomof choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, givenonly that the von Neumann universe, V, also satisfies that axiom.

12.3 Results

Some additional results of the theory are:

• Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element ofL(R).

• Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property ofBaire and the perfect set property.

• L(R) does not satisfy the axiom of uniformization or the axiom of real determinacy.

• R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R).

• While not every relation on the reals in L(R) has a uniformization in L(R), every such relation does have auniformization in L(R#).

• Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated inV[G]. Thus the theory of L(R) cannot be changed by forcing.

• L(R) satisfies AD+.

25

26 CHAPTER 12. L(R)

12.4 References• Woodin, W. Hugh (1988). “Supercompact cardinals, sets of reals, and weakly homogeneous trees”. Proceed-

ings of the National Academy of Sciences of theUnited States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

Chapter 13

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).

13.1 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

27

Chapter 14

Mouse (set theory)

In set theory, a mouse is a small model of (a fragment of) Zermelo–Fraenkel set theory with desirable properties.The exact definition depends on the context. In most cases, there is a technical definition of “premouse” and an addedcondition of iterability (referring to the existence of wellfounded iterated ultrapowers): a mouse is then an iterablepremouse. The notion of mouse generalizes the concept of a level of Gödel's constructible hierarchy while being ableto incorporate large cardinals.Mice are important ingredients of the construction of core models. The concept was isolated by Ronald Jensen in the1970s and has been used since then in core model constructions of many authors. An urban legend says that “mice”was originally a misprint for “nice”, but Jensen has denied this.

14.1 References• Dodd, A.; Jensen, R. (1981), “The core model”, Ann. Math. Logic 20 (1): 43–75, MR 0611394

• 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.

• Mitchell, William (1979), “Ramsey cardinals and constructibility”, J. Symbolic Logic 44 (2): 260–266, MR0534574

28

Chapter 15

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.

15.1 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))

15.2 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] .

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30 CHAPTER 15. SILVER MACHINE

15.3 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 16

Skolem’s paradox

In mathematical logic and philosophy, Skolem’s paradox is a seeming contradiction that arises from the downwardLöwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects ofthe theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is notan actual antinomy like Russell’s paradox, the result is typically called a paradox, and was described as a “paradoxicalstate of affairs” by Skolem (1922: p. 295).Skolem’s paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a modelthat is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence thatintuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable.Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countablesets, satisfies the first order sentence that intuitively states “there are uncountable sets”.Amathematical explanation of the paradox, showing that it is not a contradiction inmathematics, was given by Skolem(1922). Skolem’s work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic,but the result quickly came to be accepted by the mathematical community.The philosophical implications of Skolem’s paradox have received much study. One line of inquiry questions whetherit is accurate to claim that any first-order sentence actually states “there are uncountable sets”. This line of thoughtcan be extended to question whether any set is uncountable in an absolute sense. More recently, the paper “Modelsand Reality” by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of Skolem’sresult.

16.1 Background

One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets,such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set X is countableif there is a function that gives a one-to-one correspondence between X and the natural numbers, and is uncountableif there is no such correspondence function. When Zermelo proposed his axioms for set theory in 1908, he provedCantor’s theorem from them to demonstrate their strength.Löwenheim (1915) and Skolem (1920, 1923) proved the Löwenheim–Skolem theorem. The downward form of thistheorem shows that if a countable first-order axiomatisation is satisfied by any infinite structure, then the same axiomsare satisfied by some countable structure. In particular, this implies that if the first order versions of Zermelo’s axiomsof set theory are satisfiable, they are satisfiable in some countable model. The same is true of any consistent first orderaxiomatisation of set theory.

16.2 The paradoxical result and its mathematical implications

Skolem (1922) pointed out the seeming contradiction between the Löwenheim–Skolem theorem on the one hand,which implies that there is a countable model of Zermelo’s axioms, and Cantor’s theorem on the other hand, whichstates that uncountable sets exist, and which is provable from Zermelo’s axioms. “So far as I know,” Skolem writes,

31

32 CHAPTER 16. SKOLEM’S PARADOX

“no one has called attention to this peculiar and apparently paradoxical state of affairs. By virtue of the axioms wecan prove the existence of higher cardinalities... How can it be, then, that the entire domain B [a countable modelof Zermelo’s axioms] can already be enumerated by means of the finite positive integers?" (Skolem 1922, p. 295,translation by Bauer-Mengelberg)More specifically, let B be a countable model of Zermelo’s axioms. Then there is some set u in B such that B satisfiesthe first-order formula saying that u is uncountable. For example, u could be taken as the set of real numbers in B.Now, because B is countable, there are only countably many elements c such that c ∈ u according to B, because thereare only countably many elements c in B to begin with. Thus it appears that u should be countable. This is Skolem’sparadox.Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term“set” does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition ofcountability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible torecognize that a particular set u is countable, but not countable in a particular model of set theory, because there isno set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.Skolem used the term “relative” to describe this state of affairs, where the same set is included in two models of settheory, is countable in one model, and is not countable in the other model. He described this as the “most important”result in his paper. Contemporary set theorists describe concepts that do not depend on the choice of a transitivemodel as absolute. From their point of view, Skolem’s paradox simply shows that countability is not an absoluteproperty in first order logic. (Kunen 1980 p. 141; Enderton 2001 p. 152; Burgess 1977 p. 406).Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a foundationalsystem:

“I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foun-dation of mathematics that mathematicians would, for the most part, not be very much concerned withit. But in recent times I have seen to my surprise that so many mathematicians think that these axiomsof set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time hadcome for a critique.” (Ebbinghaus and van Dalen, 2000, p. 147)

16.3 Reception by the mathematical community

A central goal of early research into set theory was to find a first order axiomatisation for set theory which wascategorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem’s result showedthis is not possible, creating doubts about the use of set theory as a foundation of mathematics. It took some time forthe theory of first-order logic to be developed enough for mathematicians to understand the cause of Skolem’s result;no resolution of the paradox was widely accepted during the 1920s. Fraenkel (1928) still described the result as anantinomy:

“Neither have the books yet been closed on the antinomy, nor has agreement on its significance andpossible solution yet been reached.” (van Dalen and Ebbinghaus, 2000, p. 147).

In 1925, von Neumann presented a novel axiomatization of set theory, which developed into NBG set theory. Verymuch aware of Skolem’s 1922 paper, von Neumann investigated countable models of his axioms in detail. In hisconcluding remarks, Von Neumann comments that there is no categorical axiomatization of set theory, or any othertheory with an infinite model. Speaking of the impact of Skolem’s paradox, he wrote,

“At present we can do no more than note that we have one more reason here to entertain reservationsabout set theory and that for the time being no way of rehabilitating this theory is known."(Ebbinghausand van Dalen, 2000, p. 148)

Zermelo at first considered the Skolem paradox a hoax (van Dalen and Ebbinghaus, 2000, p. 148 ff.), and spokeagainst it starting in 1929. Skolem’s result applies only to what is now called first-order logic, but Zermelo arguedagainst the finitary metamathematics that underlie first-order logic (Kanamori 2004, p. 519 ff.). Zermelo argued thathis axioms should instead be studied in second-order logic, a setting in which Skolem’s result does not apply. Zermelopublished a second-order axiomatization in 1930 and proved several categoricity results in that context. Zermelo’s

16.4. CURRENT MATHEMATICAL OPINION 33

further work on the foundations of set theory after Skolem’s paper led to his discovery of the cumulative hierarchyand formalization of infinitary logic (van Dalen and Ebbinghaus, 2000, note 11).Fraenkel et al. (1973, pp. 303–304) explain why Skolem’s result was so surprising to set theorists in the 1920s.Gödel’s completeness theorem and the compactness theorem were not proved until 1929. These theorems illumi-nated the way that first-order logic behaves and established its finitary nature, although Gödel’s original proof of thecompleteness theorem was complicated. Leon Henkin’s alternative proof of the completeness theorem, which is nowa standard technique for constructing countable models of a consistent first-order theory, was not presented until1947. Thus, in 1922, the particular properties of first-order logic that permit Skolem’s paradox to go through werenot yet understood. It is now known that Skolem’s paradox is unique to first-order logic; if set theory is studied usinghigher-order logic with full semantics then it does not have any countable models, due to the semantics being used.

16.4 Current mathematical opinion

Current mathematical logicians do not view Skolem’s paradox as any sort of fatal flaw in set theory. Kleene (1967,p. 324) describes the result as “not a paradox in the sense of outright contradiction, but rather a kind of anomaly”.After surveying Skolem’s argument that the result is not contradictory, Kleene concludes “there is no absolute notionof countability.” Hunter (1971, p. 208) describes the contradiction as “hardly even a paradox”. Fraenkel et al. (1973,p. 304) explain that contemporary mathematicians are no more bothered by the lack of categoricity of first-ordertheories than they are bothered by the conclusion of Gödel’s incompleteness theorem that no consistent, effective, andsufficiently strong set of first-order axioms is complete.Countable models of ZF have become common tools in the study of set theory. Forcing, for example, is oftenexplained in terms of countable models. The fact that these countable models of ZF still satisfy the theorem that thereare uncountable sets is not considered a pathology; van Heijenoort (1967) describes it as “a novel and unexpectedfeature of formal systems.” (van Heijenoort 1967, p. 290)Althoughmathematicians no longer consider Skolem’s result paradoxical, the result is often discussed by philosophers.In the setting of philosophy, a merely mathematical resolution of the paradox may be less than satisfactory.

16.5 References

• Barwise, Jon (1977), “An introduction to first-order logic”, in Barwise, Jon, ed. (1982), Handbook of Math-ematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN978-0-444-86388-1

• Timothy Bays (2000). Reflections on Skolem’s Paradox (PDF) (Ph.D. thesis). UCLA Philosophy Department.

• Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972), What is mathematical logic?,London-Oxford-New York: Oxford University Press, ISBN 0-19-888087-1, Zbl 0251.02001

• Dirk Van Dalen; Heinz-Dieter Ebbinghaus (Jun 2000). “Zermelo and the Skolem Paradox”. The Bulletin ofSymbolic Logic 6 (2): 145—161.

• Dragalin, A.G. (2001), “S/s085750”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Abraham Fraenkel, Yehoshua Bar-Hillel, Azriel Levy, Dirk van Dalen (1973), Foundations of Set Theory,North-Holland.

• Henkin, L. (1950), “Completeness in the theory of types”, The Journal of Symbolic Logic 15 (2): 81–91,doi:10.2307/2266967, JSTOR 2266967.

• Kanamori, Akihiro (2004), “Zermelo and set theory”, The Bulletin of Symbolic Logic 10 (4): 487–553, doi:10.2178/bsl/1102083759,ISSN 1079-8986, MR 2136635

• Stephen Cole Kleene, (1952, 1971 with emendations, 1991 10th printing), Introduction to Metamathematics,North-Holland Publishing Company, Amsterdam NY. ISBN 0-444-10088-1. cf pages 420-432: § 75. Axiomsystems, Skolem’s paradox, the natural number sequence.

34 CHAPTER 16. SKOLEM’S PARADOX

• Stephen Cole Kleene, (1967). Mathematical Logic.

• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North-Holland,ISBN 978-0-444-85401-8

• Löwenheim, Leopold (1915), "Über Möglichkeiten im Relativkalkül” (PDF), Mathematische Annalen 76 (4):447–470, doi:10.1007/BF01458217, ISSN 0025-5831

• Moore, A.W., “Set Theory, Skolem’s Paradox and the Tractatus”, Analysis 1985: 45, doi:10.2307/3327397.

• Hilary Putnam (Sep 1980). “Models and Reality” (PDF). The Journal of Symbolic Logic 45 (3): 464—482.

• Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: SpringerScience+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

• Skolem, Thoralf (1922). “Axiomatized set theory”. Reprinted in From Frege to Gödel, van Heijenoort, 1967,in English translation by Stefan Bauer-Mengelberg, pp. 291–301.

16.6 External links• Vaughan Pratt’s celebration of his academic ancestor Skolem’s 120th birthday

• Extract from Moore’s discussion of the paradox(broken link)

Chapter 17

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.

35

Chapter 18

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.

18.1 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.

18.2 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#.

36

18.3. STATEMENTS EQUIVALENT TO EXISTENCE OF 0# 37

18.3 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#.

18.4 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.

18.5 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.

18.6 See also

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

18.7 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

38 CHAPTER 18. ZERO SHARP

• 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

18.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 39

18.8 Text and image sources, contributors, and licenses

18.8.1 Text• Axiom of constructibility Source: https://en.wikipedia.org/wiki/Axiom_of_constructibility?oldid=646125895 Contributors: Michael

Hardy, Schneelocke, Charles Matthews, Maximus Rex, Aleph4, Tobias Bergemann, Lethe, Mboverload, Barnaby dawson, AshtonBenson,Oleg Alexandrov, Penumbra2000, Masnevets, Hmonroe, Hairy Dude, Trovatore, Bota47, SmackBot, Bluebot, Cybercobra, Loadmaster,JRSpriggs, CBM, Jokes Free4Me, Spellmaster, Epsilon0, TrulyBlue, Alexbot, HOOTmag, Addbot, Lightbot, Luckas-bot, Citation bot,VladimirReshetnikov, RedBot, Chricho, ZéroBot, The Nut, ClueBot NG, Loriendrew, Blamelooseradiolabel and Anonymous: 16

• Chang’s model Source: https://en.wikipedia.org/wiki/Chang’{}s_model?oldid=621453134 Contributors: R.e.b., David Eppstein andDeltahedron

• Code (set theory) Source: https://en.wikipedia.org/wiki/Code_(set_theory)?oldid=376574787 Contributors: Charles Matthews, Giftlite,Mets501, JRSpriggs and Hans Adler

• Constructible universe Source: https://en.wikipedia.org/wiki/Constructible_universe?oldid=660903046 Contributors: Zundark, TobyBartels, Edward, Mark Foskey, Schneelocke, Charles Matthews, Dysprosia, Onebyone, Aleph4, Aetheling, Tobias Bergemann, Giftlite,Lethe, Gro-Tsen, Jason Quinn, Ben Standeven, Crisófilax, EmilJ, AshtonBenson, Aleph0~enwiki, Oleg Alexandrov, OwenX, Ryan Reich,BD2412, Rjwilmsi, Tim!, R.e.b., UkPaolo, Aatu, Trovatore, Tsca.bot, NYKevin, Rizome~enwiki, Mets501, Zero sharp, JRSpriggs, CBM,Jokes Free4Me, Gregbard, ProfStevie, Headbomb, JAnDbot, Ling.Nut, Ttwo, Terrek, Sigmundur, DH85868993, Alan U. Kennington,Hotfeba, Neithan Agarwaen, Addbot, Andrewtp21, Lightbot, Yobot, Citation bot, Citation bot 1, Obankston, WikitanvirBot, TannerSwett, Helpful Pixie Bot, PhnomPencil, IkamusumeFan, YFdyh-bot, Kephir, Leonard Huang, Mark viking and Anonymous: 19

• Core model Source: https://en.wikipedia.org/wiki/Core_model?oldid=645485084 Contributors: Dmytro, Ben Standeven, Chalst, Salixalba, Trovatore, FlashSheridan, Bluebot, Coremodel, JRSpriggs, Ntsimp, Magioladitis, AnomieBOT, BrideOfKripkenstein, DrilBot, Cn-williams, Mark viking and Anonymous: 3

• Covering lemma Source: https://en.wikipedia.org/wiki/Covering_lemma?oldid=617810862 Contributors: Zundark, Samw, Dysprosia,Dmytro, Oleg Alexandrov, Julien Tuerlinckx, R.e.b., Trovatore, FF2010, SmackBot, Silly rabbit, JRSpriggs, David Eppstein, Kope,Erik9bot, RjwilmsiBot and Anonymous: 1

• Extender (set theory) Source: https://en.wikipedia.org/wiki/Extender_(set_theory)?oldid=674666272 Contributors: Ben Standeven,Leyo, Hans Adler, Cnwilliams, Chimpionspeak, BG19bot, BattyBot and Mkoeberl

• Gödel operation Source: https://en.wikipedia.org/wiki/G%C3%B6del_operation?oldid=635363307Contributors: Michael Hardy, R.e.b.,TutterMouse and K9re11

• Innermodel Source: https://en.wikipedia.org/wiki/Inner_model?oldid=607825826Contributors: Zundark,Michael Hardy, CharlesMatthews,Giftlite, Aleph0~enwiki, R.e.b., Mathbot, Trovatore, SmackBot, Viebel, Zero sharp, JRSpriggs, Myasuda, Hans Adler, Addbot, Laaknor-Bot, Yobot, Erik9bot, VS6507, Helpful Pixie Bot, Mark viking and Anonymous: 8

• Inner model theory Source: https://en.wikipedia.org/wiki/Inner_model_theory?oldid=613805448 Contributors: TakuyaMurata, TobiasBergemann, R.e.b., Trovatore, CBM, Gregbard, Qwfp, Locobot, Mark viking and Anonymous: 1

• Jensen hierarchy Source: https://en.wikipedia.org/wiki/Jensen_hierarchy?oldid=684458070 Contributors: The Anome, Michael Hardy,Delirium, Giftlite, Ben Standeven, R.e.b., SmackBot, Zero sharp, Kope, Robin S, Chimpionspeak, Helpful Pixie Bot, ChrisGualtieri andAnonymous: 1

• L(R) Source: https://en.wikipedia.org/wiki/L(R)?oldid=635375890 Contributors: Zundark, Schneelocke, Giftlite, Waltpohl, Mysidia,Ben Standeven, Trovatore, Pgk, Bluebot, Ligulembot, Beetstra, JRSpriggs, DOI bot, Citation bot 1, Brirush and Anonymous: 2

• Minimalmodel (set theory) Source: https://en.wikipedia.org/wiki/Minimal_model_(set_theory)?oldid=661311108Contributors: Rjwilmsi,R.e.b., JRSpriggs, David Eppstein, Yobot, Citation bot, Citation bot 1 and Trappist the monk

• Mouse (set theory) Source: https://en.wikipedia.org/wiki/Mouse_(set_theory)?oldid=680138484Contributors: Tobias Bergemann, R.e.b.,Trovatore, Coremodel, CBM, Hans Adler, Erik9bot and K9re11

• Silvermachine Source: https://en.wikipedia.org/wiki/Silver_machine?oldid=625105261Contributors: Michael Hardy, CharlesMatthews,Mboverload, Barnaby dawson, RJHall, Oleg Alexandrov, Rjwilmsi, Friedfish, SmackBot, Jupix, Mets501, Kope, R'n'B, XxTimberlakexx,Kodiologist and Anonymous: 1

• Skolem’s paradox Source: https://en.wikipedia.org/wiki/Skolem’{}s_paradox?oldid=672347617 Contributors: Michael Hardy, Chinju,Graue, TakuyaMurata, Peter Damian (original account), Charles Matthews, Ruakh, Tobias Bergemann, Tosha, Giftlite, Gdm, Leibniz,Pjacobi, Phiwum, Porton, Nortexoid, Mdd, 4v4l0n42, Noosphere, Spambit, Joriki, Rjwilmsi, R.e.b., Magidin, Chris Pressey, Yurik-Bot, Joth, Trovatore, Mrwright, Curpsbot-unicodify, SmackBot, Imz, Mhss, Joshtrimble, Wvbailey, Loadmaster, Vaughan Pratt, CBM,Myasuda, Aquishix, Maproom, LordAnubisBOT, VanishedUserABC, Dmcq, CBM2, PixelBot, Oshanker, Hans Adler, Hugo Herbelin,Addbot, Unzerlegbarkeit, Lightbot, Yobot, Ptbotgourou, Maltelauridsbrigge, AnomieBOT, Citation bot, Xzungg, Sophus Bie, Citationbot 1, Tkuvho, Trappist the monk, EmausBot, WikitanvirBot, ZéroBot, Tijfo098, Helpful Pixie Bot, Deltahedron, Jochen Burghardt,Hanoch Ben-Yami and Anonymous: 27

• Statements true in L Source: https://en.wikipedia.org/wiki/Statements_true_in_L?oldid=660956623 Contributors: Chinju, Schnee-locke, Charles Matthews, Tobias Bergemann, Gene Ward Smith, Barnaby dawson, Kundor, JRSpriggs, Jokes Free4Me, R'n'B, Volons,Dexbot and Anonymous: 1

• Zero sharp Source: https://en.wikipedia.org/wiki/Zero_sharp?oldid=674929853 Contributors: Michael Hardy, TakuyaMurata, Schnee-locke, Charles Matthews, Dysprosia, Dmytro, Onebyone, Phil Boswell, Aleph4, Peak, Tobias Bergemann, Giftlite, Zeimusu, KlemenKocjancic, Barnaby dawson, Dmeranda, Ben Standeven, Gauge, EmilJ, Oleg Alexandrov, Marudubshinki, Mandarax, R.e.b., Trovatore,Leeannedy, AndrewWTaylor, SmackBot, Chris the speller, Aecea 1, Lambiam, Loadmaster, Zero sharp, JRSpriggs, Vanisaac, CBM,Headbomb, Kope, LokiClock, Yobot, Citation bot, Access Denied, BattyBot, Deltahedron, Blackbombchu and Anonymous: 9

40 CHAPTER 18. ZERO SHARP

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