Relative consistency of Set Matrix Theory with ZF

TL;DR

The paper proves the relative consistency of Set Matrix Theory (SMT) with ZF by constructing a bridge through an auxiliary theory $ZFM$, which extends $ZF$ with matrix-defining function symbols $f_{m\times n}$. It shows $ZFM$ is a definitional (conservative) extension of $ZF$, and establishes translations between SMT and $ZFM$ so that most SMT axioms (excluding epsilon and division) are mirrored as theorems in $ZFM$. A subtheory SMT$^{-}$ is analyzed, with interpretations in $ZFM$, and the relative-consistency chain SMT$^{-}$ → $ZFM$ → $ZF$ is used to conclude SMT is relatively consistent with ZF. The work also discusses the treatment of transitive sets and ordinals in SMT, addressing potential ambiguities and proposing robust definitions to maintain a coherent ordinal theory within SMT. Overall, SMT is positioned as a safe foundation for contexts where set matrices are treated as sui generis objects, provided careful handling of transitive-set notions is maintained.

Abstract

Set Matrix Theory (SMT) has been introduced in Log. Anal. 225: 59-82 (2014) as a generalization of ZF, in which matrices constructed from sets are treated as urelements, that is, as objects that are not sets but that can be elements of sets. Here we prove that SMT is relatively consistent with ZF.

Theorems & Definitions (9)

• Definition 1: Set matrices in ZF
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Corollary 7
• Proposition 8
• Proposition 9