The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

Tin Adlešić; Vedran Čačić

doi:10.18778/0138-0680.2023.25

Authors

Tin Adlešić University of Zagreb, Faculty of Teacher education
Vedran Čačić University of Zagreb, Faculty of Science https://orcid.org/0000-0003-2330-9740

DOI:

https://doi.org/10.18778/0138-0680.2023.25

Keywords:

Quine's New Foundations, cardinal multiplication, axiomatization

Abstract

In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU + Inf + AC). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (VCSP), which is a theorem of NFU + Inf + AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatic extension is NFU + Inf + AC + VCSP, which is equivalent to NFU + Inf + AC, but easier to reason about.

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