Principles of reflection in axiomatic set theory

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 Fuzzy Set Theory and Its Applications

Proof Theory, Modal Logic and Reflection Principles

Sergei Artemov Reflection vs. co-Reflection in Intuitionistic Epistemic Logic. This is joint work with Tudor Protopopescu. We outline an intuitionistic view of knowledge which maintains the Brouwer-Heyting-Kolmogorov semantics and is consistent with Williamson’s suggestion that intuitionistic knowledge is the result of verification. On this view, co-Reflection F → KF is valid since it encodes a...