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 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

Standard foundations for nonstandard analysis

1992 ◽  
Vol 57 (2) ◽  
pp. 741-748 ◽  
Author(s):  
David Ballard ◽  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Natural Place ◽  
Nonstandard Set Theory ◽  
Standard Set ◽  
The Subject ◽  
Internal Sets ◽  
Well Foundedness

In the thirty years since its invention by Abraham Robinson, nonstandard analysis has become a useful tool for research in many areas of mathematics. It seems fair to say, however, that the search for practically satisfactory foundations for the subject is not yet completed. New proposals, intended to remedy various shortcomings of older approaches, continue to be put forward. The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features. We do this by working in Zermelo-Fraenkel set theory with non-well-founded sets. It has always been clear that the axiom of regularity may fail for external sets. The previous approaches either avoid non-well-foundedness by considering only that fragment of nonstandard set theory that is well-founded (over individuals; enlargements of Robinson and Zakon [17]) or reluctantly live with it (various axiomatic nonstandard set theories). Ballard and Davidon [2] were the first to propose constructive use for non-well-foundedness in the foundations of nonstandard analysis. In the present paper we adopt a very strong anti-foundation axiom. In the resulting more or less “usual” set theory, the (to the “standard” mathematician) unfamiliar concepts of standard, external and internal sets can be defined and their requisite properties proved (rather than postulated, as is the case in axiomatic nonstandard set theories).

Standard sets in nonstandard set theory

2004 ◽  
Vol 69 (1) ◽  
pp. 165-182 ◽  
Author(s):  
Petr Andreev ◽  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Class Theory ◽  
Nonstandard Set Theory ◽  
Standard Sets

AbstractWe prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.

scholarly journals Realism, nonstandard set theory, and large cardinals

2001 ◽  
Vol 109 (1-2) ◽  
pp. 15-48 ◽  
Author(s):  
Karel Hrbacek
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Nonstandard Set Theory
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