[PPT] - ARITHMETIC, SET THEORY, AND THEIR MODELS PART ONE: END EXTENSIONS PowerPoint Presentation

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ARITHMETIC, SET THEORY, AND THEIR MODELS

PART ONE: END EXTENSIONS Ali Enayat

YOUNG SET THEORY WORKSHOP

K¨ ONIGSWINTER, MARCH 21-25, 2011

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW

(1) [Koepke-Koerwien] SO ≈ ZFC. SO = Second Order Theory of Ordinals.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW

(1) [Koepke-Koerwien] SO ≈ ZFC. SO = Second Order Theory of Ordinals. T−∞ := T\{Infinity} ∪ {¬Infinity}.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW

(1) [Koepke-Koerwien] SO ≈ ZFC. SO = Second Order Theory of Ordinals. T−∞ := T\{Infinity} ∪ {¬Infinity}. (2) [Mostowski] Z2 + Π1

∞-AC =

(Second Order Arithmetic + Choice Scheme) ≈

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW

(1) [Koepke-Koerwien] SO ≈ ZFC. SO = Second Order Theory of Ordinals. T−∞ := T\{Infinity} ∪ {¬Infinity}. (2) [Mostowski] Z2 + Π1

∞-AC =

(Second Order Arithmetic + Choice Scheme) ≈ ZFC\{Power} + V = H(ℵ1) ≈ KMC−.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW (CONT’D)

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW (CONT’D)

(3) [Ackermann, Mycielski, Kaye-Wong] ACA0 ≈ GBC−∞ + TC. PA ≈ ZF−∞ + TC.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW (CONT’D)

(3) [Ackermann, Mycielski, Kaye-Wong] ACA0 ≈ GBC−∞ + TC. PA ≈ ZF−∞ + TC. (4) [Gaifman-Dimitracopoulos] EFA (Elementary Function Arithmetic)≈ Mac−∞.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL TWINS OF ARITHMETICAL THEORIES: THE GLOBAL VIEW (CONT’D)

(3) [Ackermann, Mycielski, Kaye-Wong] ACA0 ≈ GBC−∞ + TC. PA ≈ ZF−∞ + TC. (4) [Gaifman-Dimitracopoulos] EFA (Elementary Function Arithmetic)≈ Mac−∞. (5) [Szmielew-Tarski] Robinson’s Q ≈ AST (Adjunctive Set Theory).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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FAMILIAR INTERPRETATIONS

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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FAMILIAR INTERPRETATIONS

Example A: Poincare’s interpretation of hyperbolic geometry in euclidean geometry.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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FAMILIAR INTERPRETATIONS

Example A: Poincare’s interpretation of hyperbolic geometry in euclidean geometry. Example B: Hamilton’s interpretation of ACF0 in RCF.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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FAMILIAR INTERPRETATIONS

Example A: Poincare’s interpretation of hyperbolic geometry in euclidean geometry. Example B: Hamilton’s interpretation of ACF0 in RCF. Example C: von Neumann’s interpretation of PA in ZF.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS, THE OFFICIAL DEFINITION

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS, THE OFFICIAL DEFINITION

An interpretation of a theory S in a theory T, written, I : S → T consists of a translation of each formula ϕ of S into a formula ϕI in the language of T such that (S ⊢ ϕ) = ⇒ T ⊢ ϕI.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS, THE OFFICIAL DEFINITION

An interpretation of a theory S in a theory T, written, I : S → T consists of a translation of each formula ϕ of S into a formula ϕI in the language of T such that (S ⊢ ϕ) = ⇒ T ⊢ ϕI. The translation ϕ − → ϕI is induced by the following: (a) A universe of discourse designated by a first order formula U of T; (b) A distinguished definable equivalence relation E on to interpret equality on U; (c) A T-formula ϕR(x0, · · ·, xn−1) for each n-ary relation symbol of S.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS, THE OFFICIAL DEFINITION

An interpretation of a theory S in a theory T, written, I : S → T consists of a translation of each formula ϕ of S into a formula ϕI in the language of T such that (S ⊢ ϕ) = ⇒ T ⊢ ϕI. The translation ϕ − → ϕI is induced by the following: (a) A universe of discourse designated by a first order formula U of T; (b) A distinguished definable equivalence relation E on to interpret equality on U; (c) A T-formula ϕR(x0, · · ·, xn−1) for each n-ary relation symbol of S. We write S ≤I T if S can be interpreted in T.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND MODELS

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND MODELS

In model theoretic terms: if S ≤I T, then one can uniformly interpret a model BA of S in every model A of T.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND MODELS

In model theoretic terms: if S ≤I T, then one can uniformly interpret a model BA of S in every model A of T. S and T are said to be bi-interpretable if

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND MODELS

In model theoretic terms: if S ≤I T, then one can uniformly interpret a model BA of S in every model A of T. S and T are said to be bi-interpretable if (1) T can verify that A ∼ = ABA, and

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND MODELS

In model theoretic terms: if S ≤I T, then one can uniformly interpret a model BA of S in every model A of T. S and T are said to be bi-interpretable if (1) T can verify that A ∼ = ABA, and (2) S can verify that B ∼ = BAB.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND RELATIVE CONSISTENCY

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND RELATIVE CONSISTENCY

Theorem. Suppose S and T are axiomatizable theories. Then

S ≤I T ⇒ Con(T) → Con(S).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND RELATIVE CONSISTENCY

Theorem. Suppose S and T are axiomatizable theories. Then

S ≤I T ⇒ Con(T) → Con(S). But the converse of the above can be false, e.g., for S = GB, and T = ZF.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND RELATIVE CONSISTENCY

Theorem. Suppose S and T are axiomatizable theories. Then

S ≤I T ⇒ Con(T) → Con(S). But the converse of the above can be false, e.g., for S = GB, and T = ZF. Therefore “interpretability strength”is a refinement of “consistency strength”.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND RELATIVE CONSISTENCY

Theorem. Suppose S and T are axiomatizable theories. Then

S ≤I T ⇒ Con(T) → Con(S). But the converse of the above can be false, e.g., for S = GB, and T = ZF. Therefore “interpretability strength”is a refinement of “consistency strength”.

Theorem. [Mostowski-Robinson-Tarski] If T is axiomatizable

and Q ≤I T , then T is essentially undecidable.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND SET THEORETIC INDEPENDENCE RESULTS

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND SET THEORETIC INDEPENDENCE RESULTS

Theorem. EFA ⊢ Con(ZF) → Con(ZF + V = L).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND SET THEORETIC INDEPENDENCE RESULTS

Theorem. EFA ⊢ Con(ZF) → Con(ZF + V = L).

Proof: Syntactically unwind the usual proof!

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND SET THEORETIC INDEPENDENCE RESULTS

Theorem. EFA ⊢ Con(ZF) → Con(ZF + V = L).

Proof: Syntactically unwind the usual proof! The above works since G¨

del’s L has a uniform definition

across all model of ZF, and ZF ⊢ (ZF + V = L)L.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND SET THEORETIC INDEPENDENCE RESULTS

Theorem. EFA ⊢ Con(ZF) → Con(ZF + V = L).

Proof: Syntactically unwind the usual proof! The above works since G¨

del’s L has a uniform definition

across all model of ZF, and ZF ⊢ (ZF + V = L)L.

Theorem. EFA ⊢ Con(ZF) → Con(ZF + ¬CH).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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INTERPRETATIONS AND SET THEORETIC INDEPENDENCE RESULTS

Theorem. EFA ⊢ Con(ZF) → Con(ZF + V = L).

Proof: Syntactically unwind the usual proof! The above works since G¨

del’s L has a uniform definition

across all model of ZF, and ZF ⊢ (ZF + V = L)L.

Theorem. EFA ⊢ Con(ZF) → Con(ZF + ¬CH).

Proof: Move within L and build LB, where B =c.b.a for adding ℵ2 Cohen reals; then mod out LB by the L-least ultrafilter on B.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Second order arithmetic (Z2) is a two-sorted theory; one sort for numbers, and the other sort for reals. Models of Z2 are of the form (M, A), where A ⊆ P(M), such that (1) For each X ∈ A, (M, X) | = PA(X), and (2) If X ⊆ M is parametrically definable in (M, A), then X ∈ A.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Second order arithmetic (Z2) is a two-sorted theory; one sort for numbers, and the other sort for reals. Models of Z2 are of the form (M, A), where A ⊆ P(M), such that (1) For each X ∈ A, (M, X) | = PA(X), and (2) If X ⊆ M is parametrically definable in (M, A), then X ∈ A. The Choice Scheme Π1

∞-AC consists of the universal closure

f formulae of the form

∀n ∃X ϕ(n, X) → ∃Y ∀n ϕ(n, (Y )n)

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Tanalysis := Z2 + Π1

∞-AC, and

Tset := ZFC\{Power} + V = H(ℵ1) are bi-interpretable.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Tanalysis := Z2 + Π1

∞-AC, and

Tset := ZFC\{Power} + V = H(ℵ1) are bi-interpretable. In particular, there is a canonical one-to-one correspondence between models of Tanalysis and Tset; ω-models of Tanalysis correspond to ω-models of Tset.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Tanalysis := Z2 + Π1

∞-AC, and

Tset := ZFC\{Power} + V = H(ℵ1) are bi-interpretable. In particular, there is a canonical one-to-one correspondence between models of Tanalysis and Tset; ω-models of Tanalysis correspond to ω-models of Tset. To interpret an model A Tset within a model (N∗, A) Tanalysis, one defines the notion of “suitable trees”, and an equivalence relation =∗ among suitable trees, and a binary relation ∈∗ among the equivalence classes of =∗ This yields a model A = (A, E) of Tset; where A is the set of equivalence classes of =∗ and E = ∈∗ .

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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SET THEORETICAL COUNTERPART OF SECOND ORDER ARITHMETIC-PART II

Tanalysis := Z2 + Π1

∞-AC, and

Tset := ZFC\{Power} + V = H(ℵ1) are bi-interpretable. In particular, there is a canonical one-to-one correspondence between models of Tanalysis and Tset; ω-models of Tanalysis correspond to ω-models of Tset. To interpret an model A Tset within a model (N∗, A) Tanalysis, one defines the notion of “suitable trees”, and an equivalence relation =∗ among suitable trees, and a binary relation ∈∗ among the equivalence classes of =∗ This yields a model A = (A, E) of Tset; where A is the set of equivalence classes of =∗ and E = ∈∗ . Conversely, if A is a model of Tset, then the “standard model

f second order arithmetic” in the sense of A is a model of

Tanalysis.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Theorem. [Ackermann, 1940] Let E be defined by aEb iff the

a-th component of the base-2 expansion of b is 1. Then (ω, E) ∼ = (Vω, ∈).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Theorem. [Ackermann, 1940] Let E be defined by aEb iff the

a-th component of the base-2 expansion of b is 1. Then (ω, E) ∼ = (Vω, ∈).

Theorem. PA is bi-interpretable with ZFfin.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Theorem. [Ackermann, 1940] Let E be defined by aEb iff the

a-th component of the base-2 expansion of b is 1. Then (ω, E) ∼ = (Vω, ∈).

Theorem. PA is bi-interpretable with ZFfin.

Models of ZFfin all “believe” that V = H(ℵ0).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Theorem. [Ackermann, 1940] Let E be defined by aEb iff the

a-th component of the base-2 expansion of b is 1. Then (ω, E) ∼ = (Vω, ∈).

Theorem. PA is bi-interpretable with ZFfin.

Models of ZFfin all “believe” that V = H(ℵ0).

Theorem. ACA0 is bi-interpretable with GBC−∞.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Theorem. [Ackermann, 1940] Let E be defined by aEb iff the

a-th component of the base-2 expansion of b is 1. Then (ω, E) ∼ = (Vω, ∈).

Theorem. PA is bi-interpretable with ZFfin.

Models of ZFfin all “believe” that V = H(ℵ0).

Theorem. ACA0 is bi-interpretable with GBC−∞.

Models of ACA0 are of the form (M, A), where A ⊆ P(M), such that (1) For each X ∈ A, (M, X) | = PA(X), and (2) Given A1, · · ·, An in A, if X ⊆ M is parametrically first

rder definable in (M, A1, · · ·, An), then X ∈ A.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ACKERMANN’S EPSILON RELATION

Theorem. [Ackermann, 1940] Let E be defined by aEb iff the

a-th component of the base-2 expansion of b is 1. Then (ω, E) ∼ = (Vω, ∈).

Theorem. PA is bi-interpretable with ZFfin.

Models of ZFfin all “believe” that V = H(ℵ0).

Theorem. ACA0 is bi-interpretable with GBC−∞.

Models of ACA0 are of the form (M, A), where A ⊆ P(M), such that (1) For each X ∈ A, (M, X) | = PA(X), and (2) Given A1, · · ·, An in A, if X ⊆ M is parametrically first

rder definable in (M, A1, · · ·, An), then X ∈ A.
Theorem. Z2 + Π1

∞-AC is bi-interpretable with KMC−∞.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Models of set theory are of the form M = (M, E), where E = ∈M .

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Models of set theory are of the form M = (M, E), where E = ∈M . M is standard if E is well-founded.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Models of set theory are of the form M = (M, E), where E = ∈M . M is standard if E is well-founded. M is ω-standard if (ω, <)M ∼ = (ω, <).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Models of set theory are of the form M = (M, E), where E = ∈M . M is standard if E is well-founded. M is ω-standard if (ω, <)M ∼ = (ω, <).

Proposition. Suppose M |

= ZF±∞ + TC. M is nonstandard iff (Ord, ∈)M is not well-founded.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Models of set theory are of the form M = (M, E), where E = ∈M . M is standard if E is well-founded. M is ω-standard if (ω, <)M ∼ = (ω, <).

Proposition. Suppose M |

= ZF±∞ + TC. M is nonstandard iff (Ord, ∈)M is not well-founded.

Proposition. Every M has a elementary extension that is

not ω-standard.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART I

Models of set theory are of the form M = (M, E), where E = ∈M . M is standard if E is well-founded. M is ω-standard if (ω, <)M ∼ = (ω, <).

Proposition. Suppose M |

= ZF±∞ + TC. M is nonstandard iff (Ord, ∈)M is not well-founded.

Proposition. Every M has a elementary extension that is

not ω-standard. Theorem [Keisler-Morely]. Every ω-standard countable M | = ZF has a nonstandard elementary extension that is ω-standard.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

For M = (M, E), and m ∈ M, mE := {x ∈ M : xEm}.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

For M = (M, E), and m ∈ M, mE := {x ∈ M : xEm}. X⊆ M is coded in M if X = mE for some m ∈ M.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

For M = (M, E), and m ∈ M, mE := {x ∈ M : xEm}. X⊆ M is coded in M if X = mE for some m ∈ M. Suppose M ⊆ N = (N, F) with m ∈ M. N is said to fix m if mE = mF, else N enlarges m.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

For M = (M, E), and m ∈ M, mE := {x ∈ M : xEm}. X⊆ M is coded in M if X = mE for some m ∈ M. Suppose M ⊆ N = (N, F) with m ∈ M. N is said to fix m if mE = mF, else N enlarges m. M ⊆end N, if mE = mF for every m ∈ M.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

For M = (M, E), and m ∈ M, mE := {x ∈ M : xEm}. X⊆ M is coded in M if X = mE for some m ∈ M. Suppose M ⊆ N = (N, F) with m ∈ M. N is said to fix m if mE = mF, else N enlarges m. M ⊆end N, if mE = mF for every m ∈ M. M ⊆rank N for every x ∈ N\M, and every y ∈ M, N ρ(x) > ρ(y).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODELS OF SET THEORY-PART II

For M = (M, E), and m ∈ M, mE := {x ∈ M : xEm}. X⊆ M is coded in M if X = mE for some m ∈ M. Suppose M ⊆ N = (N, F) with m ∈ M. N is said to fix m if mE = mF, else N enlarges m. M ⊆end N, if mE = mF for every m ∈ M. M ⊆rank N for every x ∈ N\M, and every y ∈ M, N ρ(x) > ρ(y). M ⊆cons N if the intersection of any parametrically definable subset of N with M is also parametrically definable in M.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODEL THEORETIC PRELIMINARIES -PART III

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODEL THEORETIC PRELIMINARIES -PART III

Proposition. (a) Rank extensions are end extensions, but not vice versa. (b) If M end N | = ZFC , then M ⊆rank N. (c) If M | = ZFfin and M cons N, then M ⊆end N.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODEL THEORETIC PRELIMINARIES -PART III

Proposition. (a) Rank extensions are end extensions, but not vice versa. (b) If M end N | = ZFC , then M ⊆rank N. (c) If M | = ZFfin and M cons N, then M ⊆end N. Theorem [Splitting Theorem]. Suppose M ≺ N where M is a model of ZF±∞ . There exist a model N ∗ such that M cofinal N ∗ end N.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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MODEL THEORETIC PRELIMINARIES -PART III

Proposition. (a) Rank extensions are end extensions, but not vice versa. (b) If M end N | = ZFC , then M ⊆rank N. (c) If M | = ZFfin and M cons N, then M ⊆end N. Theorem [Splitting Theorem]. Suppose M ≺ N where M is a model of ZF±∞ . There exist a model N ∗ such that M cofinal N ∗ end N. Theorem [E]. If M cons N | = ZFC and N fixes ωM, then M is cofinal in N.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!). Suppose κ is a cardinal in M | = ZFC.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!). Suppose κ is a cardinal in M | = ZFC. (a) M ≺κ-end N if M ≺ N and κ is end extended in the passage from M to N, i.e., N enlarges κ but N fixes every element of κE (where E =∈M).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!). Suppose κ is a cardinal in M | = ZFC. (a) M ≺κ-end N if M ≺ N and κ is end extended in the passage from M to N, i.e., N enlarges κ but N fixes every element of κE (where E =∈M). (b) M ≺κ-cons-end N if M ≺κ-end N, and the intersection of any parametrically definable (or equivalently: coded) subset of N with the ”old” elements of κ is coded in M.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!). Suppose κ is a cardinal in M | = ZFC. (a) M ≺κ-end N if M ≺ N and κ is end extended in the passage from M to N, i.e., N enlarges κ but N fixes every element of κE (where E =∈M). (b) M ≺κ-cons-end N if M ≺κ-end N, and the intersection of any parametrically definable (or equivalently: coded) subset of N with the ”old” elements of κ is coded in M. (c) An ultrafilter U over the Boolean algebra PM(κ) is said to be M-complete if for each f : κ → λ < κ in M, f is constant on some member of U.

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!). Suppose κ is a cardinal in M | = ZFC. (a) M ≺κ-end N if M ≺ N and κ is end extended in the passage from M to N, i.e., N enlarges κ but N fixes every element of κE (where E =∈M). (b) M ≺κ-cons-end N if M ≺κ-end N, and the intersection of any parametrically definable (or equivalently: coded) subset of N with the ”old” elements of κ is coded in M. (c) An ultrafilter U over the Boolean algebra PM(κ) is said to be M-complete if for each f : κ → λ < κ in M, f is constant on some member of U. (d) An M-complete ultrafilter U is M-normal if for each regressive f : κ → κ in M ∃α < κ f −1{α} ∈ U.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS

Theorem [Keisler-Tarski; Scott] κ is a measurable cardinal iff V ≺κ-end N for some N (note: κ = ω allowed!). Suppose κ is a cardinal in M | = ZFC. (a) M ≺κ-end N if M ≺ N and κ is end extended in the passage from M to N, i.e., N enlarges κ but N fixes every element of κE (where E =∈M). (b) M ≺κ-cons-end N if M ≺κ-end N, and the intersection of any parametrically definable (or equivalently: coded) subset of N with the ”old” elements of κ is coded in M. (c) An ultrafilter U over the Boolean algebra PM(κ) is said to be M-complete if for each f : κ → λ < κ in M, f is constant on some member of U. (d) An M-complete ultrafilter U is M-normal if for each regressive f : κ → κ in M ∃α < κ f −1{α} ∈ U. (e) An M-complete ultrafilter U is M-iterable if for each f : κ → κ in M {α < κ : f −1{α} ∈ U} ∈ M.

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U.

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U. (1) U is M-complete iff M ≺κ-endMU.

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U. (1) U is M-complete iff M ≺κ-endMU. (2) U is M-normal iff M ≺κ-endMU and [id]U is the least new ordinal.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U. (1) U is M-complete iff M ≺κ-endMU. (2) U is M-normal iff M ≺κ-endMU and [id]U is the least new ordinal. (3) U is M-iterable iff M ≺κ-cons-end MU .

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U. (1) U is M-complete iff M ≺κ-endMU. (2) U is M-normal iff M ≺κ-endMU and [id]U is the least new ordinal. (3) U is M-iterable iff M ≺κ-cons-end MU .

Theorem. Suppose M is a countable model of ZFC and κ is

a cardinal in M.

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U. (1) U is M-complete iff M ≺κ-endMU. (2) U is M-normal iff M ≺κ-endMU and [id]U is the least new ordinal. (3) U is M-iterable iff M ≺κ-cons-end MU .

Theorem. Suppose M is a countable model of ZFC and κ is

a cardinal in M. (1) κ is a regular cardinal in M iff there is an M-complete ultrafilter U over PM(κ).

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KAPPA ELEMENTARY END EXTENSIONS: KEISLER-MORLEY, KUNEN

Theorem Suppose M is a model of ZFC, κ is a cardinal of M, and U is a nonprincipal ultrafilter over PM(κ), and MU := Mκ/U. (1) U is M-complete iff M ≺κ-endMU. (2) U is M-normal iff M ≺κ-endMU and [id]U is the least new ordinal. (3) U is M-iterable iff M ≺κ-cons-end MU .

Theorem. Suppose M is a countable model of ZFC and κ is

a cardinal in M. (1) κ is a regular cardinal in M iff there is an M-complete ultrafilter U over PM(κ). (2) κ is weakly compact in M iff there is an M-iterable ultrafilter U over PM(κ).

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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KAPPA ELEMENTARY END EXTENSIONS, CONT’D

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KAPPA ELEMENTARY END EXTENSIONS, CONT’D

Theorem. Suppose M is a model of ZFC, κ is a regular

cardinal of M, and

PM(κ)
= ℵ0.Then M has a proper

κ-e.e.e.

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KAPPA ELEMENTARY END EXTENSIONS, CONT’D

Theorem. Suppose M is a model of ZFC, κ is a regular

cardinal of M, and

PM(κ)
= ℵ0.Then M has a proper

κ-e.e.e. Moreover, if κ is weakly compact in M, then the extension can be arranged to be κ-conservative. Theorem [Kleinberg] Suppose Mis a countable well-founded model of ZFC. κ is completely ineffable in M iff there is an ultrafilter U on PM(κ) such that U is both M- iterable and M-normal.

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KAPPA ELEMENTARY END EXTENSIONS, CONT’D

Theorem. Suppose M is a model of ZFC, κ is a regular

cardinal of M, and

PM(κ)
= ℵ0.Then M has a proper

κ-e.e.e. Moreover, if κ is weakly compact in M, then the extension can be arranged to be κ-conservative. Theorem [Kleinberg] Suppose Mis a countable well-founded model of ZFC. κ is completely ineffable in M iff there is an ultrafilter U on PM(κ) such that U is both M- iterable and M-normal. Theorem [E] There is no set Φ of first order sentences in the language {∈, κ} such that for all countable models M of ZFC, M | = Φ iff there is an ultrafilter U on PM(κ) such that U is both M-iterable and M-normal.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS

Theorem [MacDowell-Specker, Gaifman, Phillips] (a) Every model M of ZF−∞ + TC has an e.e.e N. (b) Moreover, N can be required to be a minimal and conservative extension of M.

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS

Theorem [MacDowell-Specker, Gaifman, Phillips] (a) Every model M of ZF−∞ + TC has an e.e.e N. (b) Moreover, N can be required to be a minimal and conservative extension of M. Theorem [Keisler-Morley] Every model of ZF of countable cofinality has an e.e.e of any prescribed cardinality.

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS

Theorem [MacDowell-Specker, Gaifman, Phillips] (a) Every model M of ZF−∞ + TC has an e.e.e N. (b) Moreover, N can be required to be a minimal and conservative extension of M. Theorem [Keisler-Morley] Every model of ZF of countable cofinality has an e.e.e of any prescribed cardinality. Theorem [Knight, Schmerl-Kossak, E] Every countable model of ZF−∞ + TC has continuum-many superminimal e.e.e.’s.

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ELEMENTARY END EXTENSIONS: ”BAD” NEWS

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ELEMENTARY END EXTENSIONS: ”BAD” NEWS

Theorem [Keisler-Silver] If κ is the first inaccessible cardinal, then (Vκ, ∈) has no e.e.e.

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Theorem [Keisler-Silver] If κ is the first inaccessible cardinal, then (Vκ, ∈) has no e.e.e. Theorem [Kaufmann-E] No model of ZFC has a conservative e.e.e.

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS

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ELEMENTARY END EXTENSIONS: ”BAD” NEWS

Theorem [Keisler-Silver] If κ is the first inaccessible cardinal, then (Vκ, ∈) has no e.e.e. Theorem [Kaufmann-E] No model of ZFC has a conservative e.e.e. Theorem [Kaufmann-E] Every consistent extension of ZFC has a model M of power ℵ1 such that M has no e.e.e.

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS REGAINED

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Theorem [Kaufmann-E] The following are equivalent for a consistent complete extension T of ZFC: (a) T can be expanded to a consistent theory T ∗ in an extended countable language L such that ZFC(L) ⊆ T ∗ and every model of T ∗has an e.e.e. (b) For each natural number n, T ⊢ ∃κ(κ is n-Mahlo and Vκ ≺Σn V).

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS REGAINED

Theorem [Kaufmann-E] The following are equivalent for a consistent complete extension T of ZFC: (a) T can be expanded to a consistent theory T ∗ in an extended countable language L such that ZFC(L) ⊆ T ∗ and every model of T ∗has an e.e.e. (b) For each natural number n, T ⊢ ∃κ(κ is n-Mahlo and Vκ ≺Σn V). L´ evy Scheme: Λ := {(∃κ(κ is n−Mahlo and Vκ ≺Σn V) :n < ω}

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ELEMENTARY END EXTENSIONS: ”GOOD” NEWS REGAINED

Theorem [Kaufmann-E] The following are equivalent for a consistent complete extension T of ZFC: (a) T can be expanded to a consistent theory T ∗ in an extended countable language L such that ZFC(L) ⊆ T ∗ and every model of T ∗has an e.e.e. (b) For each natural number n, T ⊢ ∃κ(κ is n-Mahlo and Vκ ≺Σn V). L´ evy Scheme: Λ := {(∃κ(κ is n−Mahlo and Vκ ≺Σn V) :n < ω} SLOGAN: ZFC + Λ is the weakest extension of ZFC that allows infinite set theory to model-theoretically catch-up with finite set theory!

Ali Enayat ARITHMETIC, SET THEORY, AND THEIR MODELS