Nonstandard Set Theory

1979 ◽  
Vol 86 (8) ◽  
pp. 659 ◽  
Author(s):  
Karel Hrbacek

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




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


2002 ◽  
Vol 67 (1) ◽  
pp. 315-325 ◽  
Author(s):  
Mauro Di Nasso

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.



2001 ◽  
Vol 66 (3) ◽  
pp. 1321-1341 ◽  
Author(s):  
P. V. Andreev ◽  
E. I. Gordon

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.





1992 ◽  
Vol 57 (2) ◽  
pp. 741-748 ◽  
Author(s):  
David Ballard ◽  
Karel Hrbacek

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).



2004 ◽  
Vol 69 (1) ◽  
pp. 165-182 ◽  
Author(s):  
Petr Andreev ◽  
Karel Hrbacek

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.



2001 ◽  
Vol 109 (1-2) ◽  
pp. 15-48 ◽  
Author(s):  
Karel Hrbacek


Sign in / Sign up

Export Citation Format

Share Document