Definability in models of set theory

1980 ◽  
Vol 45 (1) ◽  
pp. 9-19
Author(s):  
David Guaspari
Keyword(s):  
Standard Model ◽  
Unique Solution ◽  
Set Theory ◽  
Recursion Theory ◽  
Complete Characterization ◽  
Standard Part ◽  
New Year’S Eve ◽  
Single Formula ◽  
Models Of Set Theory

Call a set A of ordinals “definable” over a theory T if T is some brand of set theory and whenever A appears in the standard part of a (not necessarily standard) model of T, A is “definable”. Two kinds of “definability” are considered, for each of which is provided a complete (or almost complete) characterization of the hereditarily countable sets of ordinals “definable” over true finitely axiomatizable set theories: (1) there is a single formula ϕ such that in any model of T containing A, A is the unique solution to ϕ; (2) the defining formula is allowed to vary from model to model. (Note. The restrictions “finitely axiomatizable”, and “true” are largely for the sake of convenience: such theories provably have lots of models.)There are few allusions to what a model theorist would regard as his subject—the methods coming from recursion theory and set theory; but the treatment is intended to be intelligible to nonspecialists. The referee's criticisms have greatly improved the exposition.I would like to thank Leo Harrington for several discussions, both helpful and hapless, and especially for a clever and timely proof which rescued this project from a moribund state. (Further thanks are due to the Movshon family, as a result of whose New Year's Eve party it became clear that the only really magic formulas are Σ1 formulas.)

Omitting types: application to recursion theory

1972 ◽  
Vol 37 (1) ◽  
pp. 81-89 ◽  
Author(s):  
Thomas J. Grilliot
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Significant Contribution ◽  
Hard Core ◽  
Completeness Theorem ◽  
Recursion Theory ◽  
Omitting Types ◽  
Admissible Ordinals ◽  
Models Of Set Theory

Omitting-types theorems have been useful in model theory to construct models with special characteristics. For instance, one method of proving the ω-completeness theorem of Henkin [10] and Orey [20] involves constructing a model that omits the type {x ≠ 0, x ≠ 1, x ≠ 2,···} (i.e., {x ≠ 0, x ≠ 1, x ≠ 2,···} is not satisfiable in the model). Our purpose in this paper is to illustrate uses of omitting-types theorems in recursion theory. The Gandy-Kreisel-Tait Theorem [7] is the most well-known example. This theorem characterizes the class of hyperarithmetical sets as the intersection of all ω-models of analysis (the so-called hard core of analysis). The usual way to prove that a nonhyperarithmetical set does not belong to the hard core is to construct an ω-model of analysis that omits the type representing the set (Application 1). We will find basis results for and s — sets that are stronger than results previously known (Applications 2 and 3). The question of how far the natural hierarchy of hyperjumps extends was first settled by a forcing argument (Sacks) and subsequently by a compactness argument (Kripke, Richter). Another problem solved by a forcing argument (Sacks) and then by a compactness argument (Friedman-Jensen) was the characterization of the countable admissible ordinals as the relativized ω1's. Using omitting-types technique, we will supply a third kind of proof of these results (Applications 4 and 5). S. Simpson made a significant contribution in simplifying the proof of the latter result, with the interesting side effect that Friedman's result on ordinals in models of set theory is immediate (Application 6). One approach to abstract recursiveness and hyperarithmeticity on a countable set is to tenuously identify the set with the natural numbers. This approach is equivalent to other approaches to abstract recursion (Application 7). This last result may also be proved by a forcing method.

Uniformly defined descending sequences of degrees

1976 ◽  
Vol 41 (2) ◽  
pp. 363-367 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Unique Solution ◽  
Set Theory ◽  
Limit Point ◽  
Infinite Sequence ◽  
Linear Ordering ◽  
Natural Numbers ◽  
Linear Orderings ◽  
Nonstandard Models ◽  
Related Paper ◽  
Models Of Set Theory

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π11 sets of natural numbers are precisely those which are defined by a Σ11 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each xn+ 1 is a set in the ω-model coded by xn, and each xn+1 is the unique solution of P(xn, xn+1) for some fixed arithmetic P.

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

On the standard part of nonstandard models of set theory

1983 ◽  
Vol 48 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Menachem Magidor ◽  
Saharon Shelah ◽  
Jonathan Stavi
Keyword(s):  
Set Theory ◽  
Nonstandard Model ◽  
Standard Part ◽  
Nonstandard Models ◽  
Models Of Set Theory

AbstractWe characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

The strength of admissibility without foundation

1984 ◽  
Vol 49 (3) ◽  
pp. 867-879 ◽  
Author(s):  
Gerhard Jäger
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Constructive Mathematics ◽  
Theoretic Analysis ◽  
Second Order Arithmetic ◽  
Admissible Sets ◽  
Order Arithmetic ◽  
Image Position

The following is part of a series of papers on theories for (iterated) admissible sets (cf. [10], [11], [12], [14], [15]). Although these theories are weak subsystems of Zermelo-Fraenkel set theory, they allow one to formalize and prove a fair amount of definability theory and generalized recursion theory. Using this machinery it is in general not very hard to establish the connections between theories for admissible sets and (for example) systems of second order arithmetic. A proof-theoretic analysis of theories for admissible sets therefore provides quite a uniform and powerful framework for the proof-theoretic treatment of many systems of set theory, second order arithmetic and constructive mathematics (see [12] and [15]). The strongest result in this direction so far is the pair of proof-theoretic equivalenceswhere T0 is Feferman's system for explicit mathematics of [5] and [6], (-CA) + (BI) is the usual system of second order arithmetic with the axiom of -comprehension and bar induction and KPi is Kripke-Platek set theory with ∈-inductionfor arbitrary formulas and the additional axiom.The least standard model of KPi is L(i0) where i0 is the first recursively inaccessible ordinal.In this paper we are mainly interested in the theory KPi0 which results from KPi by severely restricting the principles of induction. Basically, complete induction on the natural numbersis allowed only for ∆0-formulas, and (IND∈) is omitted completely.

Models of set theory with more real numbers than ordinals

1974 ◽  
Vol 39 (3) ◽  
pp. 579-583 ◽  
Author(s):  
Paul E. Cohen
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Infinite Cardinal ◽  
Real Numbers ◽  
Transitive Model ◽  
Lower Case ◽  
The Axiom Of Choice ◽  
Dependent Choice ◽  
Models Of Set Theory

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

Experimental Characterization of Mechanical Behavior of Cord-Rubber Composites

1982 ◽  
Vol 10 (1) ◽  
pp. 37-54 ◽  
Author(s):  
M. Kumar ◽  
C. W. Bert
Keyword(s):  
Response Characteristic ◽  
Cord Compression ◽  
Complete Characterization ◽  
Strain Response ◽  
Strain Gages ◽  
Experimental Characterization ◽  
Rubber Composites ◽  
Stress Strain ◽  
Strain Data

Abstract Unidirectional cord-rubber specimens in the form of tensile coupons and sandwich beams were used. Using specimens with the cords oriented at 0°, 45°, and 90° to the loading direction and appropriate data reduction, we were able to obtain complete characterization for the in-plane stress-strain response of single-ply, unidirectional cord-rubber composites. All strains were measured by means of liquid mercury strain gages, for which the nonlinear strain response characteristic was obtained by calibration. Stress-strain data were obtained for the cases of both cord tension and cord compression. Materials investigated were aramid-rubber, polyester-rubber, and steel-rubber.

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