Cardinality in a paraconsistent and paracomplete set theory

TL;DR

This paper develops a theory of cardinality within a paraconsistent and paracomplete set theory, analyzing set sizes and cardinal arithmetic for inconsistent and incomplete sets.

Contribution

It introduces a novel cardinality framework in $ ext{BZFC}$, representing set sizes as a linear combination of three fundamental cardinals.

Findings

Cardinality of any set can be expressed as a linear combination of three fundamental cardinals.

Cardinal numbers form a three-dimensional space over classical cardinals.

The theory extends classical notions to inconsistent and incomplete sets.

Abstract

This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that ``$x\in A$'' is neither true nor false for some $x$). We carefully analyze what it means for two potentially incomplete or inconsistent sets to have ``the same size'', construct the corresponding cardinal numbers, and develop the basic theory of cardinal arithmetic. A surprising result is that the cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, our cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals.