There are more non-Cantorian sets than are Cantorian

TL;DR

In the NF setting, the paper analyzes how Cantorian and non-Cantorian sets compare and introduces a precise predicate-size comparison $P \prec Q$ using Frege-style cardinality $|\cdot|$ and Cantor-style injectivity $<$; it then constructs $X$, the set of surjective images of $\iota V$, and $Y=\{V\times x: x\in V\}$ to show that all Cantorian sets lie in $X$ while elements of $Y$ are non-Cantorian, with a strict inequality $|X|<|Y|$ established via Bowler’s approach. The main result is that Cantorian sets are strictly smaller than non-Cantorian ones under the criterion, providing a simple internal means to phrase and prove this abundance claim in NF. This clarifies how Cantarianity interacts with surjective image constructions and offers a framework for internal comparisons of predicate sizes in NF, potentially guiding further investigations of the Cantorian/non-Cantorian divide.

Abstract

When working in NF, [1] there is a sense that there are more non-Cantorian sets than Cantorian sets. But it is not that immediate result as one expects, since they are externally equinumerous, and the qualification "Cantorian" is not stratified and so not easy to spell internally. This account stipulates a fairly simple criterion to phrase such problems and proves that per that criterion there are more non-Cantorian sets.