Cantor-von Neumann Set-theory

In this elementary paper we establish a few novel results in set theory; their interest is wholly foundational-philosophical in motivation. We show that in Cantor-Von Neumann Set-Theory, which is a reformulation of Von Neumann’s original theory of functions and things that does not introduce ‘classes’ (let alone ‘proper classes’), developed in the 1920ies, both the Pairing Axiom and ‘half’ the Axiom of Limitation are redundant — the last result is novel. Further we show, in contrast to how things are usually done, that some theorems, notably the Pairing Axiom, can be proved without invoking the Replacement Schema (F) and the Power-Set Axiom. Also the Axiom of Choice is redundant in CVN, because it a theorem of CVN. The philosophical interest of Cantor-Von Neumann SetTheory, which is very succinctly indicated, lies in the fact that it is far better suited than Zermelo-Fraenkel Set-Theory as an axiomatisation of what Hilbert famously called Cantor’s Paradise. From Cantor one needs to jump to Von Neumann, over the heads of Zermelo and Fraenkel, and then reformulate.