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Reasonning in naive set theory (with unlimited comprehension), we derive a paradox (a formal contradiction) which can be seen as a variant of the Burali-Forti paradox.

Andrzej Kisielewicz has proposed three systems of “double extension set theory” of which we have shown two to be inconsistent in an earlier paper. Kisielewicz presented an argument that the remaining system interprets ZF, which is defective: it actually shows that the surviving possibly consistent system of double extension set theory interprets ZF with Separation and Comprehension restricted to Δ0 formulas. We show that this system does interpret ZF, using an analysis of the structure of the ordinals.

Three systems of “double extension set theory” have been proposed by Andrzej Kisielewicz in two papers. In this paper, it is shown that the two stronger systems are inconsistent, and that the third, weakest system does not admit extensionality for general sets or the use of general sets as parameters in its comprehension scheme. The parameter-free version of the comprehension principle of double extension set theory is also shown to be inconsistent with extensionality. The definitions of the systems and a self-contained exposition of their properties is given, sufficient to develop the inconsistency proofs.

In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL.

A structural (as opposed to Zadeh’s quantitative) approach to fuzziness is given, based on the operator “very”, which is added to the language of set theory together with some elementary axioms about it. Due to the axiom of foundation and to a lifting axiom, the operator is proved trivial on the cumulative hierarchy of ZF. So we have to drop either foundation or lifting. Since fuzziness concerns complemented predicates rather than sets, a class theory is needed for the very operator. And of them the Kelley-Morse (KM) theory is more appropriate for reasons of class existence. Several definable realizations of the very-operator are presented in KM−. In the last section we consider the operator “very” without the lifting axiom on classes of urelements. To each structurally fuzzy set X a traditional quantitative fuzzy set X is assigned – its quantitative representation. This way we are able partly to recover ordinary fuzzy sets from the structurally fuzzy ones.

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.

It is well known that number theory can be interpreted in the usual set theories, e.g. ZF, NF and their extensions. The problem I posed for myself was to see if, conversely, a reasonably strong set theory could be interpreted in number theory. The reason I am interested in this problem is, simply, that number theory is more basic or more concrete than set theory, and hence a more concrete foundation for mathematics. A partial solution to the problem was accomplished by WTN in [2], where it was shown that a predicative set theory could be interpreted in a natural extension of pure number theory, PN, (i.e. classical first-order Peano Arithmetic). In this paper, we go a step further by showing that a reasonably strong fragment of predicative set theory can be interpreted in PN itself. We then make an attempt to show how to develop predicative fragments of mathematics in PN. If one wishes to know what is meant by “reasonably strong” and “fragment” please read on.

The main questions considered in this paper are the consistency of a variant of a set theory with intuitionistic logic, with Brouwer's principle and the investigation of the comparative power of the Church's Thesis' variants at the set theory level.

The aim of the article is to present some of the newest meta-mathematical results concerning set theory and to discuss on this basis the problem whether and to what extent set theory can serve as a basis for mathematics.