On reduction properties

1994 ◽  
Vol 59 (3) ◽  
pp. 900-911 ◽  
Author(s):  
Hirotaka Kikyo ◽  
Akito Tsuboi
Keyword(s):  
Finite Number ◽  
Predicate Symbol ◽  
Strong Reduction ◽  
Unary Predicate ◽  
Symmetric Extension ◽  
Reduction Property ◽  
Slight Extension ◽  
Uniform Reduction

Let us consider countable languages L containing a unary predicate symbol P and L− =L\{P}. We also assume that L is relational. Then for any L-structure M, N = PM can naturally be considered as an L−-substructure of M. The main object of this paper will be the study of the following question: Under what condition does M have to be ℵ0-categorical. ℵ1-categorical, or stable if N is?Hodges and Pillay [6] proved that if M is a countable symmetric extension of N and T = Th(M) is minimal over P (they said that T is one-cardinal over P), then the total categoricity of N implies that of M. This is a solution to a problem in Ahlbrandt and Ziegler [1]. The condition that “M is a symmetric extension of N” is an interpretation of the condition “every relation on N definable in M is definable within N”. We shall give several interpretations of this phrase: They are the Ø-reduction property, the reduction property, the strong reduction property, and the uniform reduction property (Definition 1). Under the assumptions, we study the question proposed above.In §3 we treat the case that M is countable and show that if T is minimal over P and M has the strong reduction property over N, then M is ℵ0-categorical if N is (Theorem 5). This is a slight extension of the result of Hodges and Pillay mentioned above. (If M is countable and saturated, then the strong reduction property is equivalent to the condition that M will be symmetric over N if we add a finite number of appropriate constants.) A counterexample to this theorem has been obtained by Hrushovski in the case that only the Ø-reduction property is assumed. We also give a stronger result: If M has the Ø-reduction property over N and is ℵ0-categorical, M\N is infinite, and N is algebraically closed, then there is an expansion M* of M such that M* is not ℵo-categorical but M* still has the Ø-reduction property over N (Theorem 6). Moreover, we give an example such that M has the uniform reduction property over N. Th(M*) is minimal over P. N is ℵ0-categorical but M is not.

Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

An axiomatization for a class of two-cardinal models

1977 ◽  
Vol 42 (2) ◽  
pp. 174-178 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Predicate Symbol ◽  
Unary Predicate ◽  
First Order ◽  
Order Language

In this note we give a simple recursive axiomatization for the class of structures of type (ℶω ℵ0). This solves a problem of Vaught which is Problem 13 in the book [1] of Chang and Keisler. The same technique is used to get a recursive axiomatization for the class of κ-like structures where κ is strongly ω-inaccessible.Let us fix throughout some recursive first-order language L, and until further notice let us suppose that included in L is a distinguished unary predicate symbol U. For cardinals κ and λ with κ ≥ λ ≥ ℵ0, we say the structure has type (κ, λ) if card(A)= κ and card . Let K(κ, λ) be the class of all structures of type (κ, λ). For each ordinal α define 2ακby 20κ = κ, and 2ακ= ⋃ {2λ: λ = 2βκ for some β < α} when α > 0. Let Vaught proved the following theorem in [7].Theorem (Vaught). Suppose a is a sentence such that for each n < ω there are κ, λ with κ > 2λn and a model of σ of type (κ, λ). Then whenever κ ≥ λ ≥ ℵ0, the sentence σ has a model of type (κ, λ).

An axiomatics for nonstandard set theory, based on von Neumann–Bernays–Gödel Theory

2001 ◽  
Vol 66 (3) ◽  
pp. 1321-1341 ◽  
Author(s):  
P. V. Andreev ◽  
E. I. Gordon
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Predicate Symbol ◽  
Unary Predicate ◽  
Von Neumann ◽  
Class Theory ◽  
Bounded Set ◽  
Nonstandard Set Theory ◽  
Alternative Set ◽  
Transfer Principles

AbstractWe present an axiomatic framework for nonstandard analysis—the Nonstandard Class Theory (NCT) which extends von Neumann–Gödel–Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms—related to it—analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory by V. Kanovei and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets (proper subclasses of sets) in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT.

A note on interpretations of many-sorted theories

1985 ◽  
Vol 50 (2) ◽  
pp. 372-374 ◽  
Author(s):  
Julian L. Hook
Keyword(s):  
Predicate Symbol ◽  
Function Symbol ◽  
Unary Predicate

Whenever many-sorted theories are discussed in logic texts (e.g., [3, pp. 483–485]), it is fashionable to observe that every many-sorted theory can be effectively replaced by an equally powerful one-sorted theory *. The theory * contains for each sort σ of a unary predicate symbol Sσ used to indicate that an individual is of sort σ; the nonlogical axioms of * are the “translations” of those of together with axioms asserting that there is at least one individual of each sort and that all function symbols behave properly with respect to sorts. This observation suggests that perhaps many-sorted theories are no more useful than one-sorted theories. That this is not always the case has been pointed out previously [1, p. 13]. The content of this note is that can be interpretable in without * being interpretable in *.If a function symbol f in a many-sorted theory takes as its arguments n terms a1,…, an of sorts σ1, …, σn, respectively and produces a term fa1 … an of sort σ, then f is said to be of type (σ1, …, σn;σ). Likewise, a predicate symbol is of type (σ1, …, σn) if it takes as its arguments n terms of sorts σ1, …, σn. We assume that for each sort σ there is a predicate symbol = σ of type (σ, σ).

The two-cardinal problem for languages of arbitrary cardinality

2010 ◽  
Vol 75 (3) ◽  
pp. 785-801
Author(s):  
Luis Miguel ◽  
Villegas Silva
Keyword(s):  
Predicate Symbol ◽  
Unary Predicate ◽  
Transfer Theorem ◽  
First Order ◽  
Order Language

AbstractLet ℒ be a first-order language of cardinality κ++ with a distinguished unary predicate symbol U. In this paper we prove, working on L, the two cardinal transfer theorem (κ+,κ) ⇒ (κ++, κ+) for this language. This problem was posed by Chang and Keisler more than twenty years ago.

Partition properties of M-ultrafilters and ideals

1987 ◽  
Vol 52 (4) ◽  
pp. 897-907
Author(s):  
Joji Takahashi
Keyword(s):  
Set Theory ◽  
Predicate Symbol ◽  
Infinite Cardinal ◽  
List Type ◽  
Weakly Compact ◽  
Unary Predicate ◽  
Transitive Model ◽  
Basic Set ◽  
Uncountable Cardinal ◽  
The Axiom Of Choice

As is well known, the following are equivalent for any uniform ultrafilter U on an uncountable cardinal:(i) U is selective;(ii) U → ;(iii) U → .In §1 of this paper, we consider this result in terms of M-ultrafilters (Definition 1.1), where M is a transitive model of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). We define the partition properties and for M-ultrafilters (Definition 1.3), and characterize those M-ultrafilters that possess these properties (Theorem 1.5) so that the result mentioned at the beginning is subsumed as the special case that M is V, the universe of all sets. It turns out that the two properties have to be handled separately, and that, besides selectivity, we need to formulate additional conditions (Definition 1.4). The extra conditions become superfluous when M = V because they are then trivially satisfied. One of them is nothing new; it is none other than Kunen's iterability-of-ultrapowers condition.In §2, we obtain characterizations of the partition properties I+ → and I+ → (Definition 2.3) of uniform ideals I on an infinite cardinal κ (Theorem 2.6). This is done by applying the main results of §1 to the canonical -ultrafilter in the Boolean-valued model constructed from the completion of the quotient algebra P(κ)/I. They are related to certain known characterizations of weakly compact and of Ramsey cardinals.Our basic set theory is ZFC. In §1, it has to be supplemented by a new unary predicate symbol M and new nonlogical axioms that make M look like a transitive model of ZFC.

Indescribable cardinals and elementary embeddings

1991 ◽  
Vol 56 (2) ◽  
pp. 439-457 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Set Theory ◽  
Predicate Symbol ◽  
Large Cardinal ◽  
Unary Predicate ◽  
Finite Sequences ◽  
Elementary Embeddings ◽  
Reflection Principles ◽  
The One ◽  
Image Position ◽  
The Universe

Indescribability is closely related to the reflection principles of Zermelo-Fränkel set theory. In this axiomatic setting the universe of all sets stratifies into a natural cumulative hierarchy (Vα: α ϵ On) such that any formula of the language for set theory that holds in the universe already holds in the restricted universe of all sets obtained by some stage.The axioms of ZF prove the existence of many ordinals α such that this reflection scheme holds in the world Vα. Hanf and Scott noticed that one arrives at a large cardinal notion if the reflecting formulas are allowed to contain second order free variables to which one assigns subsets of Vα. For a given collection Ω of formulas in the ϵ language of set theory with higher type variables and a unary predicate symbol they define an ordinal α to be Ω indescribable if for all sentences Φ in Ω and A ⊆ VαSince a sufficient coding apparatus is available, this definition is (for the classes of formulas that we are going to consider) equivalent to the one that one obtains by allowing finite sequences of relations over Vα, some of which are possibly k-ary. We will be interested mainly in certain standardized classes of formulas: Let (, respectively) denote the class of all formulas in the language introduced above whose prenex normal form has n alternating blocks of quantifiers of type m (i.e. (m + 1)th order) starting with ∃ (∀, respectively) and no quantifiers of type greater than m. In Hanf and Scott [1961] it is shown that in ZFC, indescribability is equivalent to inaccessibility and indescribability coincides with weak compactness.

Asymptotic conditional probabilities: The non-unary case

1996 ◽  
Vol 61 (1) ◽  
pp. 250-276 ◽  
Author(s):  
Adam J. Grove ◽  
Joseph Y. Halpern ◽  
Daphne Koller
Keyword(s):  
Lower Bounds ◽  
Predicate Symbol ◽  
Upper And Lower Bounds ◽  
Degrees Of Belief ◽  
Conditional Probabilities ◽  
Unary Predicate ◽  
First Order

AbstractMotivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

The decidability of hindley's axioms for strong reduction

1967 ◽  
Vol 32 (2) ◽  
pp. 237-239 ◽  
Author(s):  
Bruce Lercher
Keyword(s):  
Finite Number ◽  
Infinite System ◽  
Combinatory Logic ◽  
Strong Reduction

In his paper [3] Hindley shows that strong reduction in combinatory logic (see [1] for the basic discussion of this notion) cannot be axiomatized with a finite number of axiom schemes, and then he presents an infinite system of axiom schemes for strong reduction. Hindley lists one axiom and six axiom schemes, together with a method (clause (viii) below) for generating further axiom schemes from these. The results of this note are that different applications of clause (viii) yield different axiom schemes, and that the property of being an axiom is decidable.

ON A QUESTION OF KRAJEWSKI’S

2019 ◽  
Vol 84 (1) ◽  
pp. 343-358 ◽  
Author(s):  
FEDOR PAKHOMOV ◽  
ALBERT VISSER
Keyword(s):  
Predicate Symbol ◽  
Finite Extension ◽  
Conservative Extension ◽  
Unary Predicate ◽  
Recursively Enumerable ◽  
Image Position ◽  
Conservative Extensions

AbstractIn this paper, we study finitely axiomatizable conservative extensions of a theory U in the case where U is recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.Consider a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extension α of U in the expanded language that is conservative over U, there is a conservative extension β of U in the expanded language, such that $\alpha \vdash \beta$ and $\beta \not \vdash \alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U instead of conservative extensions. Moreover, the result is preserved when we replace $\dashv$ as ordering on the finitely axiomatized extensions in the expanded language by a relevant kind of interpretability, to wit interpretability that identically translates the symbols of the U-language.We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.

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