Eine Note zum System PC

Motto: Funktoren- und Axio-meneinsparung ist mitunter am falschen Platz.

Abstract

Corresponding to a graduation of the comprehension predicateC there arise two systems PC₁ and PC₂. PC₁ represents constructive mathematics whereas PC₂ as a restricted enlargement of PC₁ enables one to deal with full set theory. Unnatural definitions likeKuratowski's ordered pair and identification of 0 and$\not 0$ are avoided. In PC₁ the postulate introduced for ordered pairs holds not only for sets but for classes generally. The comprehension condition in PC₁ implicitly contains a set axiom for at most denumerable calsses and for ordered pairs. By introduction of the concept “ordered self-pair” the real numbers are interpreted as self-pairs of (non denumerable) equivalence classes of concentrated ratioal sequences and they can be shown to be sets in PC₁.