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 !-model Has no least !-model RCA0 X WKL0 X ACA0 X ATR0 X 1 Π1-CA0 X Z2 X Which subsystems of second-order arithmetic have a smallest !-model? K Williams (CUNY) Minimal models of Kelley{Morse set theory BEST 2016 2 / 18 Which subsystems of second-order arithmetic have a smallest !-model? Has a least !-model Has no least !-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 ' with only set quantifiers fx : '(x; P)g 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 ' with only set quantifiers fx : '(x; P)g 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 '(x; Y ; P), then there is a class Z so that '(x; Zx ; P) for every x, where Zx is the slice Zx = fy :(x; y) 2 Zg: 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 '(x; Y ; P), then there is a class Z so that '(x; Zx ; P) for every x, where Zx is the slice Zx = fy :(x; y) 2 Zg: 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 = f0; 2g. 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 = f0; 2g. 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 = f0; 2g. 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 = f0; 2g. K Williams (CUNY) Minimal models of Kelley{Morse set theory BEST 2016 9 / 18 Unrolling the second-order model X W M () + − (M; X ) j= KM W j= 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.