Authors:Janusz CZELAKOWSKI

Title:Towards the Algebraization of Set Theory. Set-theoretic Domains

Pages:157-178

File:bibtex

Abstract ( + )

In this paper an algebraic approach to set theory is outlined. Models of set theory are traditionally viewed as pairs (A,e), where A is a set (or a class), and e is a binary relation on A. In the algebraic approach models are defined as quadruples **A** = (A,<,&,0), where the reduct (A,<,0) is a partially ordered set (class) with zero and & is a binary operation defined on A. Intuitively, the elements of A are sets in the sense of **A** (**A**-sets, for short). < is the inclusion relation in the sense of **A** and 0 is the empty **A**-set. The operation & plays a crucial role in this approach. If a,b in A, then a&b is the **A**-set obtained by adjoining the **A**-set b as a new element to the **A**-set a. Thus & corresponds to the set theoretic operation which, when performed on sets X and Y, yields the set X&Y := Xu{Y}. The in-relation is then defined in terms of & by the condition: Y in X if and only if X & Y = X. In the algebraic set-up the operation &, and not the in-relation, is a primitive notion and it is characterized by a number of simple and plausible conditions.

Let L be the language having in its vocabulary a binary predicate <, a binary operation symbol &, and a constant symbol 0 as the only non-logical symbols. In the paper an axiom system for & is presented in L. Moreover, taking into account the infinitistic character of models of set theory, some further axioms are provided, e.g. in the form of plausible induction principles. These infinitistic axioms are formulated as second-order conditions but elementary counterparts of these axioms (in the form of workable schemes in L) can also be investigated. Some of the known axioms of set theory as Regularity or Ch