ARITHMETIC, SET THEORY, AND THEIR MODELS PART TWO: ENDOMORPHISMS - PowerPoint PPT Presentation

ARITHMETIC, SET THEORY, AND THEIR MODELS PART TWO: ENDOMORPHISMS Ali Enayat YOUNG SET THEORY WORKSHOP K¨ ONIGSWINTER, MARCH 21-25, 2011 Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS Suppose M = ( M , E ) is a non ω -standard model of ZF ±∞ , and c ∈ M . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS Suppose M = ( M , E ) is a non ω -standard model of ZF ±∞ , and c ∈ M . Recall c E := { x ∈ M : xEc } . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS Suppose M = ( M , E ) is a non ω -standard model of ZF ±∞ , and c ∈ M . Recall c E := { x ∈ M : xEc } . SSy ( M ) := { c E ∩ ω : c ∈ M } = the standard system of M . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS Suppose M = ( M , E ) is a non ω -standard model of ZF ±∞ , and c ∈ M . Recall c E := { x ∈ M : xEc } . SSy ( M ) := { c E ∩ ω : c ∈ M } = the standard system of M . A family A ⊆ P ( ω ) is a Scott set if A is a Boolean algebra closed under Turing reducibility which satisfies the property “every infinite subtree of 2 <ω has an infinite branch”. Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS Suppose M = ( M , E ) is a non ω -standard model of ZF ±∞ , and c ∈ M . Recall c E := { x ∈ M : xEc } . SSy ( M ) := { c E ∩ ω : c ∈ M } = the standard system of M . A family A ⊆ P ( ω ) is a Scott set if A is a Boolean algebra closed under Turing reducibility which satisfies the property “every infinite subtree of 2 <ω has an infinite branch”. Theorem . [Scott] = ZF ±∞ . (a) SSy ( M ) is a Scott set for every M | (b) If A is a countable Scott set, then A can be realized as SSy ( M ) for some model of ZF ±∞ . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Theorem . [Ehrenfeucht, Knight-Nadel] In (b) above |A| = ℵ 0 can be relaxed to |A| ≤ ℵ 1 . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Theorem . [Ehrenfeucht, Knight-Nadel] In (b) above |A| = ℵ 0 can be relaxed to |A| ≤ ℵ 1 . Corollary. Under CH, A is a Scott set iff A can be realized as SSy ( M ) for some model of ZF ±∞ . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Theorem . [Ehrenfeucht, Knight-Nadel] In (b) above |A| = ℵ 0 can be relaxed to |A| ≤ ℵ 1 . Corollary. Under CH, A is a Scott set iff A can be realized as SSy ( M ) for some model of ZF ±∞ . Scott Set Problem . Is every Scott set of the form SSy ( M ) for some model of ZF ±∞ ? Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Theorem . [Ehrenfeucht, Knight-Nadel] In (b) above |A| = ℵ 0 can be relaxed to |A| ≤ ℵ 1 . Corollary. Under CH, A is a Scott set iff A can be realized as SSy ( M ) for some model of ZF ±∞ . Scott Set Problem . Is every Scott set of the form SSy ( M ) for some model of ZF ±∞ ? Kanovei’s Problem . Is there a Borel model of ZF ±∞ such that SSy ( M ) = P ( ω )? Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Theorem . [Ehrenfeucht, Knight-Nadel] In (b) above |A| = ℵ 0 can be relaxed to |A| ≤ ℵ 1 . Corollary. Under CH, A is a Scott set iff A can be realized as SSy ( M ) for some model of ZF ±∞ . Scott Set Problem . Is every Scott set of the form SSy ( M ) for some model of ZF ±∞ ? Kanovei’s Problem . Is there a Borel model of ZF ±∞ such that SSy ( M ) = P ( ω )? Theorem [Gitman]. (ZFC + PFA) Suppose A ⊆ P ( ω ) is arithmetically closed and A / fin is proper. Then A is the standard system of some model of ZF ±∞ . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

STANDARD SYSTEMS, CONT’D Theorem . [Ehrenfeucht, Knight-Nadel] In (b) above |A| = ℵ 0 can be relaxed to |A| ≤ ℵ 1 . Corollary. Under CH, A is a Scott set iff A can be realized as SSy ( M ) for some model of ZF ±∞ . Scott Set Problem . Is every Scott set of the form SSy ( M ) for some model of ZF ±∞ ? Kanovei’s Problem . Is there a Borel model of ZF ±∞ such that SSy ( M ) = P ( ω )? Theorem [Gitman]. (ZFC + PFA) Suppose A ⊆ P ( ω ) is arithmetically closed and A / fin is proper. Then A is the standard system of some model of ZF ±∞ . Theorem [E, Shelah] There exists A ⊆ P ( ω ) that is is arithmetically closed and A / fin is proper; indeed A can be arranged to be Borel. Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

RECURSIVE SATURATION Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

RECURSIVE SATURATION Proposition. M is recursively saturated iff (1) M is not ≺ M for cofinally many α ∈ Ord M . ω -standard, and (2) V M α Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

RECURSIVE SATURATION Proposition. M is recursively saturated iff (1) M is not ≺ M for cofinally many α ∈ Ord M . ω -standard, and (2) V M α Theorem. [Ehrenfeucht-Jensen] The isomorphism type of a countable recursively saturated model M of arithmetic is determined by the following two invariants (1) Th ( M ) and (2) SSy ( M ) . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

RECURSIVE SATURATION Proposition. M is recursively saturated iff (1) M is not ≺ M for cofinally many α ∈ Ord M . ω -standard, and (2) V M α Theorem. [Ehrenfeucht-Jensen] The isomorphism type of a countable recursively saturated model M of arithmetic is determined by the following two invariants (1) Th ( M ) and (2) SSy ( M ) . (1) Recursively saturated models are homogeneous , i.e., if ( M , a 1 , · · · , a n ) ≡ ( M , b 1 , · · · , b n ), then for every c ∈ M there is d ∈ M such that ( M , a 1 , · · · , a n , c ) ≡ ( M , b 1 , · · · , b n , d ) . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

RECURSIVE SATURATION Proposition. M is recursively saturated iff (1) M is not ≺ M for cofinally many α ∈ Ord M . ω -standard, and (2) V M α Theorem. [Ehrenfeucht-Jensen] The isomorphism type of a countable recursively saturated model M of arithmetic is determined by the following two invariants (1) Th ( M ) and (2) SSy ( M ) . (1) Recursively saturated models are homogeneous , i.e., if ( M , a 1 , · · · , a n ) ≡ ( M , b 1 , · · · , b n ), then for every c ∈ M there is d ∈ M such that ( M , a 1 , · · · , a n , c ) ≡ ( M , b 1 , · · · , b n , d ) . (2) The set of n -types that are coded in a recursively saturated model of arithmetic are precisely those finitely satisfiable types whose G¨ odel numbers are coded in SSy( M ). Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

RECURSIVE SATURATION Proposition. M is recursively saturated iff (1) M is not ≺ M for cofinally many α ∈ Ord M . ω -standard, and (2) V M α Theorem. [Ehrenfeucht-Jensen] The isomorphism type of a countable recursively saturated model M of arithmetic is determined by the following two invariants (1) Th ( M ) and (2) SSy ( M ) . (1) Recursively saturated models are homogeneous , i.e., if ( M , a 1 , · · · , a n ) ≡ ( M , b 1 , · · · , b n ), then for every c ∈ M there is d ∈ M such that ( M , a 1 , · · · , a n , c ) ≡ ( M , b 1 , · · · , b n , d ) . (2) The set of n -types that are coded in a recursively saturated model of arithmetic are precisely those finitely satisfiable types whose G¨ odel numbers are coded in SSy( M ). (3) Any two countable homogeneous models that satisfy the same set of types are isomorphic. This is established by a back-and-forth argument. Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

FRIEDMAN’S SELF-EMBEDDING THEOREM Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

FRIEDMAN’S SELF-EMBEDDING THEOREM Theorem. [Friedman] Every countable nonstandard model = ZF ±∞ is isomorphic to a proper rank initial segment of M | itself. Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

FRIEDMAN’S SELF-EMBEDDING THEOREM Theorem. [Friedman] Every countable nonstandard model = ZF ±∞ is isomorphic to a proper rank initial segment of M | itself. Proof (for the non ω -standard case) . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

FRIEDMAN’S SELF-EMBEDDING THEOREM Theorem. [Friedman] Every countable nonstandard model = ZF ±∞ is isomorphic to a proper rank initial segment of M | itself. Proof (for the non ω -standard case) . V M is recursively saturated for every α ∈ Ord M . α Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

FRIEDMAN’S SELF-EMBEDDING THEOREM Theorem. [Friedman] Every countable nonstandard model = ZF ±∞ is isomorphic to a proper rank initial segment of M | itself. Proof (for the non ω -standard case) . V M is recursively saturated for every α ∈ Ord M . α Fix c ∈ ω M \ ω and for each α ∈ Ord M , consider Th ≤ c ( V M α ). Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

FRIEDMAN’S SELF-EMBEDDING THEOREM Theorem. [Friedman] Every countable nonstandard model = ZF ±∞ is isomorphic to a proper rank initial segment of M | itself. Proof (for the non ω -standard case) . V M is recursively saturated for every α ∈ Ord M . α Fix c ∈ ω M \ ω and for each α ∈ Ord M , consider Th ≤ c ( V M α ). By Replacement M ∃ α 0 ∈ Ord M such that M satisfies: α ∈ Ord M : Th ≤ c ( V M α )) = Th ≤ c ( V M � � α 0 ) is unbounded in Ord . Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS