Abstract: This chapter reviews some recent results reached in the metamathematical investigation of set theory and discusses their relevance to the problems of the foundations of mathematics. There are several essentially different notions of set, which are equally admissible as the intuitive basis for set theory. The nature of sets is very important for the foundations of mathematics. In set theory, all sets belong to the universe of the theory as well as all relations, which can be defined in a set. It is expressed by the power set axiom whose use is essential in the proof that all relations with a given field form a set. There are other more powerful constructions, which lead from a set to another more comprehensive set. The existential assumptions are formulated and accepted in set theory. These assumptions are known as "axioms of infinity." There are two general principles that allow formulating infinitely many such axioms: (1) the principle of transition from potential to actual infinity and (2) the principle of existence of singular sets. The chapter explores whether these axioms are relevant for more conservative portions of set theory, which deal with sets of limited powers. It also concentrates on the problem of characterizing true existential or conditionally existential statements concerning sets of integers.
Publication Year: 1967
Publication Date: 1967-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 49
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