Minimal models of Kelley–Morse set theory

Kameryn J Williams

CUNY Graduate Center

Boise Extravaganza in Set Theory 2016 June 15

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 1 / 18 Has a least ω-model Has no least ω-model RCA0 X WKL0 X ACA0 X ATR0 X 1 Π1-CA0 X Z2 X

Which subsystems of second-order arithmetic have a smallest ω-model?

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 2 / 18 Which subsystems of second-order arithmetic have a smallest ω-model?

Has a least ω-model Has no least ω-model RCA0 X WKL0 X ACA0 X ATR0 X 1 Π1-CA0 X Z2 X

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 2 / 18 Definition (M, X ) is transitive if X is transitive or, equivalently, if M is transitive, where M is the first-order part and X ⊆ P(M) is the second-order part.

Definition (M, X ) is the least transitive model of T if X is contained in any transitive model of T .

An analogous question for set theory

Question Which second-order set theories have a least transitive model?

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 3 / 18 An analogous question for set theory

Question Which second-order set theories have a least transitive model?

Definition (M, X ) is transitive if X is transitive or, equivalently, if M is transitive, where M is the first-order part and X ⊆ P(M) is the second-order part.

Definition (M, X ) is the least transitive model of T if X is contained in any transitive model of T .

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 3 / 18 Theorem (Folklore) Every countable model of ZFC can be extended to a model of GBC without adding any new sets. (Force to add a global well-order, close off under definability.)

G¨odel–Bernays set theory with global choice

The axioms of GBC: ZFC for sets. Extensionality for classes: classes with the same members are equal. Class Replacement: if F is a class function and a is a set then F 00a is a set. Comprehension: for formulae ϕ with only set quantifiers {x : ϕ(x, P)} is a class. Global Choice: there is a bijection V → Ord.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 4 / 18 G¨odel–Bernays set theory with global choice

The axioms of GBC: ZFC for sets. Extensionality for classes: classes with the same members are equal. Class Replacement: if F is a class function and a is a set then F 00a is a set. Comprehension: for formulae ϕ with only set quantifiers {x : ϕ(x, P)} is a class. Global Choice: there is a bijection V → Ord.

Theorem (Folklore) Every countable model of ZFC can be extended to a model of GBC without adding any new sets. (Force to add a global well-order, close off under definability.)

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 4 / 18 Theorem (Shepherdson) There is a least transitive model of GBC.

A classical result

Theorem (Shepherdson 1953, Cohen 1963) There is a least transitive model of ZFC.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 5 / 18 A classical result

Theorem (Shepherdson 1953, Cohen 1963) There is a least transitive model of ZFC.

Theorem (Shepherdson) There is a least transitive model of GBC.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 5 / 18 Theorem (Folklore) KM proves there is a truth predicate for first-order truth. In particular, KM proves Con(ZFC). (Show that KM proves that transfinite recursion can be done on class-sized things.

Define truth via the usual (class-sized) recursion.)

Stronger second-order set theories

Kelley–Morse set theory (KM) is like GBC, but the Comprehension schema is strengthened to allow formulae with class quantifiers.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 6 / 18 Stronger second-order set theories

Kelley–Morse set theory (KM) is like GBC, but the Comprehension schema is strengthened to allow formulae with class quantifiers. Theorem (Folklore) KM proves there is a truth predicate for first-order truth. In particular, KM proves Con(ZFC). (Show that KM proves that transfinite recursion can be done on class-sized things.

Define truth via the usual (class-sized) recursion.)

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 6 / 18 Fact + If κ is inaccessible then (Vκ, Vκ+1) is a model of KM .

Stronger second-order set theories

KM+ is KM plus the Class Collection axiom schema: If for every set x there is a class Y so that ϕ(x, Y , P), then there is a class Z so that ϕ(x, Zx , P) for every x, where Zx is the slice

Zx = {y :(x, y) ∈ Z}.

Z Zx

x

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 7 / 18 Stronger second-order set theories

KM+ is KM plus the Class Collection axiom schema: If for every set x there is a class Y so that ϕ(x, Y , P), then there is a class Z so that ϕ(x, Zx , P) for every x, where Zx is the slice

Zx = {y :(x, y) ∈ Z}.

Z Zx

x

Fact + If κ is inaccessible then (Vκ, Vκ+1) is a model of KM .

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 7 / 18 L in the second-order part

We can build L inside the second-order part of a model of KM, keeping the same sets. Theorem The classes of any model of KM can be shrunk to produce a model of KM+ with the same sets.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 8 / 18 + − Producing a KM model from a ZFCI model is easy. − For the other direction, represent sets for the ZFCI model by classes which are well-founded extensional binary relations with a top element. Mod out by isomorphism.

na

n2

n1

n0 represents the set a = {0, 2}.

Second-order set theory is first-order set theory in disguise

Theorem + − − KM and ZFCI are bi-interpretable, where ZFCI is ZFC − Powerset plus the assertion that there is a largest cardinal, which is inaccessible.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 9 / 18 − For the other direction, represent sets for the ZFCI model by classes which are well-founded extensional binary relations with a top element. Mod out by isomorphism.

na

n2

n1

n0 represents the set a = {0, 2}.

Second-order set theory is first-order set theory in disguise

Theorem + − − KM and ZFCI are bi-interpretable, where ZFCI is ZFC − Powerset plus the assertion that there is a largest cardinal, which is inaccessible.

+ − Producing a KM model from a ZFCI model is easy.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 9 / 18 na

n2

n1

n0 represents the set a = {0, 2}.

Second-order set theory is first-order set theory in disguise

Theorem + − − KM and ZFCI are bi-interpretable, where ZFCI is ZFC − Powerset plus the assertion that there is a largest cardinal, which is inaccessible.

+ − Producing a KM model from a ZFCI model is easy. − For the other direction, represent sets for the ZFCI model by classes which are well-founded extensional binary relations with a top element. Mod out by isomorphism.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 9 / 18 Second-order set theory is first-order set theory in disguise

Theorem + − − KM and ZFCI are bi-interpretable, where ZFCI is ZFC − Powerset plus the assertion that there is a largest cardinal, which is inaccessible.

+ − Producing a KM model from a ZFCI model is easy. − For the other direction, represent sets for the ZFCI model by classes which are well-founded extensional binary relations with a top element. Mod out by isomorphism.

na

n2

n1

n0 represents the set a = {0, 2}.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 9 / 18 Unrolling the second-order model

X W

M ⇐⇒

+ − (M, X ) |= KM W |= ZFCI

If (M, X ) = (Vκ, Vκ+1), for κ inaccessible, then W = Hκ+ .

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 10 / 18 Proof Sketch: Suppose otherwise that (M, X ) is the least transitive model of KM. (M, X ) must be a countable model of KM+.

M

The main theorem

Theorem (W.) There is no least transitive model of KM.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 11 / 18 (M, X ) must be a countable model of KM+.

M

The main theorem

Theorem (W.) There is no least transitive model of KM. Proof Sketch: Suppose otherwise that (M, X ) is the least transitive model of KM.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 11 / 18 M

The main theorem

Theorem (W.) There is no least transitive model of KM. Proof Sketch: Suppose otherwise that (M, X ) is the least transitive model of KM. (M, X ) must be a countable model of KM+.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 11 / 18 The main theorem

Theorem (W.) There is no least transitive model of KM. Proof Sketch: Suppose otherwise that (M, X ) is the least transitive model of KM. (M, X ) must be a countable model of KM+.

X

M

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 11 / 18 The main theorem

Theorem (W.) There is no least transitive model of KM. Proof Sketch: Suppose otherwise that (M, X ) is the least transitive model of KM. (M, X ) must be a countable model of KM+.

W

M

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 11 / 18 A detour through admissible sets

Theorem (H. Friedman 1973) A is countable, admissible (= transitive model of KP).

T is an LA theory which is Σ1-definable in A. T has an admissible model W ⊇ A. Then, there is U |= T + KP so that wfp(U) ⊇ A and Ordwfp(U) = OrdA.

W

A

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 12 / 18 A detour through admissible sets

Theorem (H. Friedman 1973) A is countable, admissible (= transitive model of KP).

T is an LA theory which is Σ1-definable in A. T has an admissible model W ⊇ A. Then, there is U |= T + KP so that wfp(U) ⊇ A and Ordwfp(U) = OrdA.

U

A

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 12 / 18 M

Back to the main theorem

Suppose that our (M, X ) is correct about well-foundedness.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 13 / 18 Back to the main theorem

Suppose that our (M, X ) is correct about well-foundedness.

W

M

W is transitive.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 13 / 18 Back to the main theorem

Suppose that our (M, X ) is correct about well-foundedness.

W

A

M

Find A ∈ W admissible with M ∈ A.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 13 / 18 Back to the main theorem

Suppose that our (M, X ) is correct about well-foundedness.

− U |= ZFCI

A

M

Apply Friedman’s theorem to A.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 13 / 18 Back to the main theorem

Suppose that our (M, X ) is correct about well-foundedness.

Y

M

− + Turn the ZFCI model into a KM model (M, Y).

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 13 / 18 Back to the main theorem

Suppose that our (M, X ) is correct about well-foundedness.

Y

M

X 6⊆ Y because Y doesn’t have any element representing OrdA.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 13 / 18 But it might be that W has no admissible sets containing M in its well-founded part. Thus, Friedman’s theorem must be generalized to include the ill-founded case. (Use Barwise’s notion of the admissible cover.) Also, we have to be more careful when arguing that X 6⊆ Y.

What if (M, X ) is wrong about well-foundedness?

If A is in the well-founded part of W , the same argument works.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 14 / 18 Thus, Friedman’s theorem must be generalized to include the ill-founded case. (Use Barwise’s notion of the admissible cover.) Also, we have to be more careful when arguing that X 6⊆ Y.

What if (M, X ) is wrong about well-foundedness?

If A is in the well-founded part of W , the same argument works. But it might be that W has no admissible sets containing M in its well-founded part.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 14 / 18 Also, we have to be more careful when arguing that X 6⊆ Y.

What if (M, X ) is wrong about well-foundedness?

If A is in the well-founded part of W , the same argument works. But it might be that W has no admissible sets containing M in its well-founded part. Thus, Friedman’s theorem must be generalized to include the ill-founded case. (Use Barwise’s notion of the admissible cover.)

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 14 / 18 What if (M, X ) is wrong about well-foundedness?

If A is in the well-founded part of W , the same argument works. But it might be that W has no admissible sets containing M in its well-founded part. Thus, Friedman’s theorem must be generalized to include the ill-founded case. (Use Barwise’s notion of the admissible cover.) Also, we have to be more careful when arguing that X 6⊆ Y.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 14 / 18 What if (M, X ) is wrong about well-foundedness?

If A is in the well-founded part of W , the same argument works. But it might be that W has no admissible sets containing M in its well-founded part. Thus, Friedman’s theorem must be generalized to include the ill-founded case. (Use Barwise’s notion of the admissible cover.) Also, we have to be more careful when arguing that X 6⊆ Y.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 14 / 18 Definition M |= ZFC. Say X ⊆ P(M) is a KM-realization for M if (M, X ) is a model of KM.

Theorem (W.) M a countable model of ZFC. There is no least KM-realization for M.

Theorem (W.) M a countable model of ZFC. There is a least GBC-realization for M if and only if M has a definable global well-order if and only if M is a model of ∃x V = HOD({x}).

Some corollaries

Theorem (W.) For any countable set r, there is no least transitive model of KM containing r.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 15 / 18 Theorem (W.) M a countable model of ZFC. There is a least GBC-realization for M if and only if M has a definable global well-order if and only if M is a model of ∃x V = HOD({x}).

Some corollaries

Theorem (W.) For any countable set r, there is no least transitive model of KM containing r.

Definition M |= ZFC. Say X ⊆ P(M) is a KM-realization for M if (M, X ) is a model of KM.

Theorem (W.) M a countable model of ZFC. There is no least KM-realization for M.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 15 / 18 if and only if M is a model of ∃x V = HOD({x}).

Some corollaries

Theorem (W.) For any countable set r, there is no least transitive model of KM containing r.

Definition M |= ZFC. Say X ⊆ P(M) is a KM-realization for M if (M, X ) is a model of KM.

Theorem (W.) M a countable model of ZFC. There is no least KM-realization for M.

Theorem (W.) M a countable model of ZFC. There is a least GBC-realization for M if and only if M has a definable global well-order

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 15 / 18 Some corollaries

Theorem (W.) For any countable set r, there is no least transitive model of KM containing r.

Definition M |= ZFC. Say X ⊆ P(M) is a KM-realization for M if (M, X ) is a model of KM.

Theorem (W.) M a countable model of ZFC. There is no least KM-realization for M.

Theorem (W.) M a countable model of ZFC. There is a least GBC-realization for M if and only if M has a definable global well-order if and only if M is a model of ∃x V = HOD({x}).

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 15 / 18 Recent work by others

Theorem (Antos & S. Friedman) For any real r there is a least β-model of KM+ containing r.

Definition (M, X ) is a β-model if it is correct about well-foundedness.

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 16 / 18 What about the uncountable case? Can an uncountable model of ZFC have a least KM-realization? A least GBC-realization? Fact There are uncountable models of ZFC without any GBC-realizations.

(Consider M |= ZFC + ∀x V 6= HOD({x}) which is rather classless.)

What if we ask for minimal, rather than least models?

Some open questions

What about the theory GBC + ETR? (ETR asserts that transfinite recursion for first-order

properties can be done along all class-sized well-founded relations. This theory is the set theoretic analogue of ATR0

from second-order arithmetic.)

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 17 / 18 Fact There are uncountable models of ZFC without any GBC-realizations.

(Consider M |= ZFC + ∀x V 6= HOD({x}) which is rather classless.)

What if we ask for minimal, rather than least models?

Some open questions

What about the theory GBC + ETR? (ETR asserts that transfinite recursion for first-order

properties can be done along all class-sized well-founded relations. This theory is the set theoretic analogue of ATR0

from second-order arithmetic.) What about the uncountable case? Can an uncountable model of ZFC have a least KM-realization? A least GBC-realization?

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 17 / 18 What if we ask for minimal, rather than least models?

Some open questions

What about the theory GBC + ETR? (ETR asserts that transfinite recursion for first-order

properties can be done along all class-sized well-founded relations. This theory is the set theoretic analogue of ATR0

from second-order arithmetic.) What about the uncountable case? Can an uncountable model of ZFC have a least KM-realization? A least GBC-realization? Fact There are uncountable models of ZFC without any GBC-realizations.

(Consider M |= ZFC + ∀x V 6= HOD({x}) which is rather classless.)

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 17 / 18 Some open questions

What about the theory GBC + ETR? (ETR asserts that transfinite recursion for first-order

properties can be done along all class-sized well-founded relations. This theory is the set theoretic analogue of ATR0

from second-order arithmetic.) What about the uncountable case? Can an uncountable model of ZFC have a least KM-realization? A least GBC-realization? Fact There are uncountable models of ZFC without any GBC-realizations.

(Consider M |= ZFC + ∀x V 6= HOD({x}) which is rather classless.)

What if we ask for minimal, rather than least models?

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 17 / 18 Thank you!

Some references: Carolin Antos & Sy-David Friedman. Hyperclass forcing in Morse-Kelley class theory. submitted Harvey Friedman. Countable models of set theories. In A.R.D. Mathias & H. Rogers, editors, Cambridge Summer School in Mathematical Logic, pages 539–573. New York, Springer-Verlag, 1973. Stephen Simpson. Subsystems of Second Order Arithmetic. New York, Springer-Verlag, 1999. Kameryn J Williams. Minimal models of second-order set theories. in preparation

K Williams (CUNY) Minimal models of Kelley–Morse set theory BEST 2016 18 / 18