Extension of standard models of ZFC to models of Nelson’s nonstandard set theory IST

1999 ◽  
Vol 66 (2) ◽  
pp. 160-166
Author(s):  
V. G. Kanovei ◽  
M. Reeken
Keyword(s):  
Set Theory ◽  
Nonstandard Set Theory ◽  
Standard Models

Nonstandard Set Theory

1979 ◽  
Vol 86 (8) ◽  
pp. 659 ◽  
Author(s):  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Nonstandard Set Theory

A nonstandard set theory in the $\displaystyle\in$ -language

2000 ◽  
Vol 39 (6) ◽  
pp. 403-416 ◽  
Author(s):  
Vladimir Kanovei ◽  
Michael Reeken
Keyword(s):  
Set Theory ◽  
Nonstandard Set Theory

scholarly journals Исчисление касательных и вокруг

2018 ◽  
Author(s):  
A.G. Kusraev ◽  
S.S. Kutateladze
Keyword(s):  
Set Theory ◽  
Variational Analysis ◽  
Mathematical Theory ◽  
Local Analysis ◽  
Modern Approach ◽  
Standard Models ◽  
Models Of Set Theory

Optimization is the choice of what is most preferable. Geometry and local analysis of nonsmooth objects are needed for variational analysis which embraces optimization. These involved admissible directions and tangents as the limiting positions of the former. The calculus of tangents is one of the main techniques of optimization. Calculus reduces forecast to numbers, which is scalarization in modern parlance. Spontaneous solutions are often labile and rarely optimal. Thus, optimization as well as calculus of tangents deals with inequality, scalarization and stability. The purpose of this article is to give an overview of the modern approach to this range of questions based on non-standard models of set theory. A model of a mathematical theory is usually called nonstandard if the membership within the model has interpretation different from that of set theory. In the recent decades much research is done into the nonstandard methods located at the junctions of analysis and logic. This area requires the study of some new opportunities of modeling that open broad vistas for consideration and solution of various theoretical and applied problems.

scholarly journals An elementary variant of nonstandard set theory

1987 ◽  
Vol 63 (5) ◽  
pp. 165-166
Author(s):  
Moto-o Kinoshita
Keyword(s):  
Set Theory ◽  
Nonstandard Set Theory

Nonstandard set Theory

1979 ◽  
Vol 86 (8) ◽  
pp. 659-677 ◽  
Author(s):  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Nonstandard Set Theory

scholarly journals An axiomatic presentation of the nonstandard methods in mathematics

2002 ◽  
Vol 67 (1) ◽  
pp. 315-325 ◽  
Author(s):  
Mauro Di Nasso
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Strong Form ◽  
Conservative Extension ◽  
Nonstandard Set Theory

AbstractA nonstandard set theory *ZFC is proposed that axiomatizes the nonstandard embedding *. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. *ZFC is a conservative extension of ZFC.

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.

scholarly journals Axiom of Choice in nonstandard set theory

2012 ◽  
Author(s):  
Hrbacek
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Nonstandard Set Theory

The model of set theory generated by countably many generic reals

1981 ◽  
Vol 46 (4) ◽  
pp. 732-752 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Minimal Model ◽  
The Other ◽  
Power Set ◽  
Standard Models ◽  
Weak Axioms Of Choice

AbstractAdjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF model including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it.

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