A contribution to Gödel's axiomatic set theory, III

Axiomatic set theory

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using, say, Venn diagrams. The intuitive approach silently assumes that all objects in the universe of discourse satisfying any defining condition form a set. This assumption gives rise to antinomies, the simplest and best known of which being Russell's paradox. Axiomatic set theory was orig...

Axiomatic Set Theory

Lecture 1, 07.03.: We made a review of the material covered in Chapter I of [3], up to Theorem I.9.11 (Transfinite Recursion on Well-founded Relations). Lecture 2, 14.03.: We discussed the notion of a rank, as well as the Mostowski collapsing function material corresponding to Section 9 of [3]. Lecture 3, 04.04.: We discussed hereditarily transitive sets, the DownwardLöwenheim-Skolem-Tarksi The...

Axiomatic Set Theory in memoriam

In the last hundred-odd years, set theory has been studied mainly as axiomatized mathematical theory. This axiomatic approach to set theory was launched by Zermelo (1908b; see also Ebbinghaus 2007). Under Hilbert‘s influence, he presented in 1908 a set of axioms for set theory. His main objective can be said to have been to remove the uncertainties from the set-theoretical foundations of mathem...

Axiomatic theory of spectrum III. — Semiregularities

The notion of regularity in a Banach algebra was introduced and studied in [KM] and [MM]. A non-empty subset R of a unital Banach algebra A is called a regularity if it satisfies the following two conditions: (i) if a ∈ A and n ∈ N, then a ∈ R ⇔ a ∈ R, (ii) if a, b, c, d are mutually commuting elements of A satisfying ac + bd = 1A then ab ∈ R ⇔ a, b ∈ R. The axioms of regularities are weak enou...

Axiomatic Fuzzy Set Theory and Its Applications