Nonstandard Models of Arithmetic and Set Theory

2004 ◽  
Keyword(s):  
Set Theory ◽  
Nonstandard Models ◽  
Models Of Arithmetic

James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - James H. Schmerl. Automorphism groups of models of Peano arithmetic. The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264. - James H. Schmerl. Diversity in substructures. Nonstandard models of arithmetic and set theory. edited by A. Enayat and R. Kossak, Contemporary Mathematics, vol. 361, American Mathematical Societey (2004), pp. 45–161. - James H. Schmerl. Generic automorphisms and graph coloring. Discrete Mathematics, vol. 291 (2005), no. 1–3, pp. 235–242. - James H. Schmerl. Nondiversity in substructures. The Journal of Symbolic Logic, vol. 73 (2008), no. 1, pp. 193–211.

2009 ◽  
Vol 15 (2) ◽  
pp. 222-227
Author(s):  
Roman Kossak
Keyword(s):  
Set Theory ◽  
Graph Coloring ◽  
Automorphism Groups ◽  
Peano Arithmetic ◽  
Discrete Mathematics ◽  
Symbolic Logic ◽  
Nonstandard Models ◽  
Recursively Saturated ◽  
University Of Connecticut ◽  
Models Of Arithmetic

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

Set theoretic properties of Loeb measure

1990 ◽  
Vol 55 (3) ◽  
pp. 1022-1036 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Zero Set ◽  
Zero Sets ◽  
Martin’S Axiom ◽  
Basic Set ◽  
Nonstandard Models ◽  
The Continuum

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

Disturbing arithmetic

1985 ◽  
Vol 50 (2) ◽  
pp. 375-379 ◽  
Author(s):  
Thomas J. Grilliot
Keyword(s):  
Fine Structure ◽  
Prime Number ◽  
Long Range ◽  
Set Theory ◽  
New Model ◽  
New Models ◽  
Models Of Arithmetic ◽  
Two Alternatives

One long-range objective of logic is to find models of arithmetic with noteworthy properties, perhaps properties that imply some long-standing number theoretic conjectures. In areas of mathematics such as algebra or set theory, new models are often made by extending old models, that is, by adjoining new elements to already existing models. Usually the extension retains most of the characteristics of the old model with at least one exception that makes the new model interesting. However, such a scheme is difficult in the area of arithmetic. Many interesting properties of the fine structure of arithmetic are diophantine and hence unchangeable in extensions. For instance, one cannot change a prime number into a composite one by adjoining new elements.One could possibly get around this diophantine difficulty in one of two ways. One way is to change the usual language of addition and multiplication to an equivalent language that does not transmit so much information to extensions. For instance, multiplication is definable from the squaring function, as one sees from the identity 2xy = (x + y)2 − x2 − y2, and the squaring function in turn is definable either from the unary square predicate (as one sees from the fact that n = m2 if n and n + 2m + 1 are successive squares) or from the divisor relation (as one sees from the fact that n = m2 if n is the smallest number such that m divides n and m + 1 divides n + m). Either of these two alternatives to multiplication might make for interesting extensions.

Omega-inconsistency without cuts and nonstandard models

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Andreas Fjellstad
Keyword(s):  
Deontic Logic ◽  
Sequent Calculus ◽  
Standard Deontic Logic ◽  
Nonstandard Models ◽  
Deontic Modality ◽  
The Relationship ◽  
Labelled Sequent Calculus ◽  
Models Of Arithmetic

This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee (1985) for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency and shows thus, pace Cobreros et al.(2013), that the result in McGee (1985) does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic.

Löwenheim–Skolem theorems and nonstandard models

2018 ◽  
Author(s):  
W.D. Hart
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Nonstandard Models

Sometimes we specify a structure by giving a description and counting anything that satisfies the description as just another model of it. But at other times we start from a conception we try to articulate, and then our articulation may fail to pin down what we had in mind. Sets seem to have had such a fate. For millennia sets lay fallow in logic, but when cultivated by mathematics in the nineteenth century, they seemed to bear both a foundation and a theory of the infinite. The paradoxes of set theory seemed to threaten this promise. With an eye to proving freedom from paradox, versions of set theory were articulated rigorously. But around 1920, Löwenheim and Skolem proved that no such formalized set theory can come out true only in the hugely infinite world it seemed to reveal, for if it is true in such a world, it will also be true in a world of the smallest infinite size. (Versions of this remain true even if we augment the standard expressive devices used to formalize set theory.) But then, Skolem inferred, we cannot articulate sets determinately enough for them to constitute a firm foundation for mathematics.

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).

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