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<p>ZF is undoubtedly the most widely accepted foundational theory for mathematics. However, ZF has two properties that already by the founding fathers of axiomatic set theory were considered pathological: ZF has infinitely many axioms, and ZF has a countable model. Recently, in Axioms 10(2):119 (2021), a nonclassical first-order theory <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathfrak {T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">T</mi> </math></EquationSource> </InlineEquation> of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF but that lacks these two pathological properties. Here we prove that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak {T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">T</mi> </math></EquationSource> </InlineEquation> is relatively consistent with ZF. We conclude that this is a concrete step towards showing that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathfrak {T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">T</mi> </math></EquationSource> </InlineEquation> is an advancement in the foundations of mathematics.</p>

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Relative Consistency of a Finite Nonclassical Theory Incorporating ZF and Category Theory with ZF

  • 期刊论文

  • 发表日期：

    2026年5月4日

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  • Marcoen J. T. F. Cabbolet,
  • Adrian R. D. Mathias

摘要

ZF is undoubtedly the most widely accepted foundational theory for mathematics. However, ZF has two properties that already by the founding fathers of axiomatic set theory were considered pathological: ZF has infinitely many axioms, and ZF has a countable model. Recently, in Axioms 10(2):119 (2021), a nonclassical first-order theory \(\mathfrak {T}\) T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF but that lacks these two pathological properties. Here we prove that \(\mathfrak {T}\) T is relatively consistent with ZF. We conclude that this is a concrete step towards showing that \(\mathfrak {T}\) T is an advancement in the foundations of mathematics.