The Hanf number of stationary logic.

Let L be a first order language containing a binary relation symbol <.Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).

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Model theory under the axiom of determinateness

Journal of Symbolic Logic ◽

10.2307/2274328 ◽

1985 ◽

Vol 50 (3) ◽

pp. 773-780

Author(s):

Mitchell Spector

Keyword(s):

Model Theory ◽

Axiom Of Choice ◽

Order Logic ◽

First Order Logic ◽

Technical Innovation ◽

First Order ◽

Partition Relations ◽

Hanf Number ◽

The Axiom Of Choice

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

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Types omitted in uncountable models of arithmetic

Journal of Symbolic Logic ◽

10.2307/2272157 ◽

1975 ◽

Vol 40 (3) ◽

pp. 317-320 ◽

Cited By ~ 3

Author(s):

Julia F. Knight

Keyword(s):

Completeness Theorem ◽

Elementary Extension ◽

Ramsey's Theorem ◽

Definable Set ◽

Ramsey’S Theorem ◽

Hanf Number ◽

Definable Sets ◽

Omitting Types ◽

Infinite Set ◽

Models Of Arithmetic

In [4] it is shown that if the structure omits a type Σ, and Σ is complete with respect to Th(), then there is a proper elementary extension of which omits Σ. This result is extended in the present paper. It is shown that Th() has models omitting Σ in all infinite powers.A type is a countable set of formulas with just the variable ν occurring free. A structure is said to omit the type Σ if no element of satisfies all of the formulas of Σ. A type Σ, in the same language as a theory T, is said to be complete with respect to T if (1) T ∪ Σ is consistent, and (2) for every formula φ(ν) of the language of T (with just ν free), either φ or ¬φ is in Σ.The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is . It is shown that there is a model of Th() which omits Σ and contains an infinite set of indiscernibles. Where Morley used the Erdös-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here.The “omitting types” version of the ω-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the ω-completeness theorem followed from the fact that, in , any infinite, definable set can be split into two infinite, definable sets.

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On existence proofs of Hanf numbers

Journal of Symbolic Logic ◽

10.2307/2272646 ◽

1974 ◽

Vol 39 (2) ◽

pp. 318-324 ◽

Cited By ~ 2

Author(s):

Harvey Friedman

Keyword(s):

Existence Proofs ◽

Axiom Scheme ◽

Hanf Number ◽

Full Separation ◽

Open Question

This paper refines some results of Barwise [1] as well as answering the open question posed at the end of [1] about the Hanf number of positively. We conclude by showing that the existence of a Hanf bound for cannot be proved in the natural formally intuitionistic set theories with bounded predicates decidable of [3], [4] and [5].All notation not explained below is taken from [1]. In the Appendix, we give the axioms of ZF0, ZF1, and T in full. We remark that an important point about the axiom of foundation was not emphasized in [1]. This axiom was intended to be the axiom scheme (∀x)((∀y ∈ x)(A(y)) → A(x)) → (∀x)(A(x)), where y does not occur in A = A(x), instead of the more customary (∀x)(∀y)(y ∈ x → (∃z ∈ x)(∀w ∈ z) (w ∉ x)). This is of no consequence in the presence of full separation, but is vital when considering ZF0 and the T below, for with the customary form of foundation, these cannot even prove the existence of Rω+ω.In [1], a proof of the following is sketched.

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Hanf number for Scott sentences of computable structures

Archive for Mathematical Logic ◽

10.1007/s00153-018-0615-6 ◽

2018 ◽

Vol 57 (7-8) ◽

pp. 889-907

Author(s):

S. S. Goncharov ◽

J. F. Knight ◽

I. Souldatos

Keyword(s):

Computable Structures ◽

Hanf Number ◽

Scott Sentences

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A Hanf number for saturation and omission: the superstable case

Mathematical Logic Quarterly ◽

10.1002/malq.201300022 ◽

2014 ◽

Vol 60 (6) ◽

pp. 437-443 ◽

Cited By ~ 2

Author(s):

John T. Baldwin ◽

Saharon Shelah

Keyword(s):

Hanf Number

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Omitting types in set theory and arithmetic

Journal of Symbolic Logic ◽

10.1017/s0022481200051719 ◽

1976 ◽

Vol 41 (1) ◽

pp. 25-32 ◽

Cited By ~ 1

Author(s):

Julia F. Knight

Keyword(s):

Set Theory ◽

Conservative Extension ◽

Alternate Proof ◽

Hanf Number ◽

Omitting Types ◽

Infinite Set ◽

Models Of Set Theory

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

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Vopěnka's principle and compact logics

Journal of Symbolic Logic ◽

10.2307/2273786 ◽

1985 ◽

Vol 50 (1) ◽

pp. 42-48 ◽

Cited By ~ 4

Author(s):

J. A. Makowsky

Keyword(s):

Finitely Generated ◽

Compact Cardinal ◽

Hanf Number ◽

Vopěnka’S Principle

AbstractWe study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

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