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Approximating Cartesian Closed Categories in NF-Style Set Theories

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Abstract

I criticize, but uphold the conclusion of, an argument by McLarty to the effect that New Foundations style set theories don’t form a suitable foundation for category theory. McLarty’s argument is from the fact that Set and Cat are not Cartesian closed in NF-style set theories. I point out that these categories do still have a property approximating Cartesian closure, making McLarty’s argument not conclusive. After considering and attempting to address other problems with developing category theory in NF-style set theories, I conclude that NF-style set theories are not a good foundation for category theory, because of numerous limitations introduced by their stratification restrictions.

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Notes

  1. There are natural mathematical statements which are not provable in NF-style set theories, such as that given any set X the map x↦{x} defined on elements of X exists. But I know of no examples of mathematical theorems — substantive statements about a non-set-theoretic subject matter — which are not provable in a suitable NF-style set theory when the concepts used in the theorem are (if necessary, re-)defined in a way that respects the relative typing discipline of these theories.

  2. Namely, (a, b)=(c, d)⇔a = c∧b = d.

  3. When we write f:A→B⊧U, this is shorthand for \(A \overset {f}{\longrightarrow } B \models U\).

References

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  3. Jensen, R.B. (1968). On the consistency of a slight(?) modification of Quine’s New Foundations. Synthese, 19(1/2).

  4. McLarty, C. (1992). Failure of Cartesian closedness in NF. The Journal of Symbolic Logic, 57(2), 555–556.

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Authors and Affiliations

  1. University of Connecticut, 101 Manchester Hall, 355 Mansfield Rd., Storrs, CT, 06269-1054, USA

    Morgan Thomas

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  1. Morgan Thomas
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Correspondence to Morgan Thomas.

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Thomas, M. Approximating Cartesian Closed Categories in NF-Style Set Theories. J Philos Logic 47, 143–160 (2018). https://doi.org/10.1007/s10992-017-9425-2

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  • DOI: https://doi.org/10.1007/s10992-017-9425-2

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