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Tangled Type Theory was introduced by Randall Holmes in [3] as a new way of approaching the consistency problem for $\mathrm {NF}$. Although the task of finding models for this theory is far from trivial (considering it is equiconsistent with $\mathrm {NF}$), ways of constructing models for certain fragments of it have been discovered. In this article, we present a simpler way of constructing models of predicative Tangled Type Theory and consequently of predicative $\mathrm {NF}$. In these new models of predicative $\mathrm {NF}$, the universe is well-orderable and equinumerous to the set of singletons.

This paper consists in the formulation of a novel categorical set theory, MLCat, which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic (denoted with an “I” on the left) or classical NF, with atoms (denoted with a “U” on the right) or not:

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CANTORIAN MODELS OF PREDICATIVE $\mathrm {NF}$

ALGEBRAIC NEW FOUNDATIONS

NFSI IS NOT INCLUDED IN NF3

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3 results

CANTORIAN MODELS OF PREDICATIVE $\mathrm {NF}$

ALGEBRAIC NEW FOUNDATIONS

NFSI IS NOT INCLUDED IN NF3