Minimal models of second-order set theories
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1 Minimal models of second-order set theories Kameryn J Williams CUNY Graduate Center Set Theory Day 2016 March 11 K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 1 / 17
2 A classical result Theorem (Shepherdson, Cohen) There is a least transitive model of ZFC. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 2 / 17
3 Gödel Bernays set theory with global choice Models of GBC consist of sets (x, y, z,...) and classes (X, Y, Z,...). For a model (M, X ) we will always assume that X P(M). The axioms: 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 a is a set. Comprehension for formulae 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 second-order set theories 2016 March 11 3 / 17
4 Gödel Bernays set theory with global choice Models of GBC consist of sets (x, y, z,...) and classes (X, Y, Z,...). For a model (M, X ) we will always assume that X P(M). The axioms: 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 a is a set. Comprehension for formulae 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 second-order set theories 2016 March 11 3 / 17
5 The same classical result Theorem (Shepherdson) There is a least transitive model of GBC. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 4 / 17
6 The same classical result Theorem (Shepherdson) There is a least transitive model of GBC. Definition (M, X ) is transitive if X is transitive or, equivalently, if M is transitive. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 4 / 17
7 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 second-order set theories 2016 March 11 5 / 17
8 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). K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 5 / 17
9 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, Z x, P) for every x, where Z x is the slice Z x = {y : (x, y) Z}. Z x Z x K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 6 / 17
10 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, Z x, P) for every x, where Z x is the slice Z x = {y : (x, y) Z}. Z x Z x Fact If κ is inaccessible then (V κ, V κ+1 ) is a model of KM +. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 6 / 17
11 Second-order set theory is first-order set theory in disguise Theorem KM + and ZFC I are bi-interpretable, where ZFC I is ZFC Powerset plus the assertion that there is a largest cardinal, which is inaccessible. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 7 / 17
12 Second-order set theory is first-order set theory in disguise Theorem KM + and ZFC I are bi-interpretable, where ZFC I is ZFC Powerset plus the assertion that there is a largest cardinal, which is inaccessible. Producing a KM + model from a ZFC I model is easy. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 7 / 17
13 Second-order set theory is first-order set theory in disguise Theorem KM + and ZFC I are bi-interpretable, where ZFC I is ZFC Powerset plus the assertion that there is a largest cardinal, which is inaccessible. Producing a KM + model from a ZFC I model is easy. For the other direction, represent sets for the ZFC I model by classes which are well-founded extensional binary relations with a top element. Mod out by isomorphism. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 7 / 17
14 Second-order set theory is first-order set theory in disguise Theorem KM + and ZFC I are bi-interpretable, where ZFC I is ZFC Powerset plus the assertion that there is a largest cardinal, which is inaccessible. Producing a KM + model from a ZFC I model is easy. For the other direction, represent sets for the ZFC I model by classes which are well-founded extensional binary relations with a top element. Mod out by isomorphism. n 2 n a n 1 n 0 represents the set a = {0, 2}. K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 7 / 17
15 Unrolling the second-order model X M W (M, X ) = KM + W = ZFC I K Williams (CUNY) Minimal models of second-order set theories 2016 March 11 8 / 17
16 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 second-order set theories 2016 March 11 9 / 17
17 The main theorem Theorem (W.) There is no least transitive model of KM. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
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. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
19 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 second-order set theories 2016 March / 17
20 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 second-order set theories 2016 March / 17
21 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 second-order set theories 2016 March / 17
22 A detour through a theorem of Harvey Friedman Theorem (H. Friedman) A is countable, admissible (= transitive model of KP). T is an L A theory which is Σ 1 -definable in A. T has an admissible model W A. Then, there is U = T + KP so that wfp(u) A and Ord wfp(u) = Ord A. W A K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
23 A detour through a theorem of Harvey Friedman Theorem (H. Friedman) A is countable, admissible (= transitive model of KP). T is an L A theory which is Σ 1 -definable in A. T has an admissible model W A. Then, there is U = T + KP so that wfp(u) A and Ord wfp(u) = Ord A. U A K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
24 Back to the main theorem Suppose that our (M, X ) is correct about well-foundedness. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
25 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 second-order set theories 2016 March / 17
26 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 second-order set theories 2016 March / 17
27 Back to the main theorem Suppose that our (M, X ) is correct about well-foundedness. U = ZFC I A M Apply Friedman s theorem to A. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
28 Back to the main theorem Suppose that our (M, X ) is correct about well-foundedness. Y M Turn the ZFC I model into a KM + model (M, Y). K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
29 Back to the main theorem Suppose that our (M, X ) is correct about well-foundedness. Y M X Y because Y doesn t have any element representing Ord A. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
30 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 second-order set theories 2016 March / 17
31 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 second-order set theories 2016 March / 17
32 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 second-order set theories 2016 March / 17
33 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 Y. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
34 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 Y. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
35 Some corollaries Theorem (W.) For any real r, there is no least transitive model of KM containing r. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
36 Some corollaries Theorem (W.) For any real 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 second-order set theories 2016 March / 17
37 Some corollaries Theorem (W.) For any real 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 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 second-order set theories 2016 March / 17
38 Some corollaries Theorem (W.) For any real 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 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 second-order set theories 2016 March / 17
39 Recent work by others Theorem (C. Antos & S. Friedman) For any real r there is a least β-model of KM + + DC containing r. Definition (M, X ) is a β-model if it is correct about well-foundedness. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
40 Some open questions 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 second-order set theories 2016 March / 17
41 Some open questions 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 HOD(x) which is rather classless. K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
42 Some open questions 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 HOD(x) which is rather classless. What if we ask for minimal, rather than least models? K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
43 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 New York, Springer-Verlag, Kameryn J Williams. Minimal models of second-order set theories. in preparation K Williams (CUNY) Minimal models of second-order set theories 2016 March / 17
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