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A few years ago I was reading an entry in the Stanford Encyclopedia of Philosophy related to set theory and I stumbled across a statement along these lines:

There is a set theory where full comprehension is allowed, and this set theory tolerates a degree of contradiction.

I am trying to find this SEP entry now to find out what this set theory is called. Neither Google nor SEP itself (even with its own special search engine) has yielded any results.

Does anybody know what that set theory in the SEP article was (or might have been)? The theory referred to was an axiomatic system like ZFC or NFU, not naive set theory (which matches the description).

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  • Hartry Field etc, Prospect for a Naive Theory of Classes, Notre Dame Journal of Formal Logic (2017). Following Paul Ross directions to Brady.
    – Mauro ALLEGRANZA
    Commented Feb 5, 2022 at 16:05
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    It's in SEP, Inconsistent Mathematics:"A number of people including da Costa (1974), Brady (1971, 1989), Priest, Routley, & Norman (1989, pp. 152, 498), considered it preferable to retain the full power of the natural comprehension principle (every predicate determines a set), and tolerate a degree of inconsistency in set theory".
    – Conifold
    Commented Feb 5, 2022 at 23:12
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    Btw, one can have something like full comprehension even without inconsistency if one stratifies bound variables as in Quine’s New Foundations.
    – Conifold
    Commented Feb 5, 2022 at 23:38

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In general, what you are referring to are paraconsistent set theories. These are versions of set theory in which the underlying logic lacks the principle of explosion, and in some cases tolerates true contradictions. These are able to accommodate an unrestricted principle of comprehension and a universal set.

Some of the people who have published in this area are R.T. Brady, Newton da Costa, Graham Priest, and Greg Restall. See for example,

Zach Weber, Transfinite Numbers in Paraconsistent Set Theory, The Review of Symbolic Logic, Volume 3, March 2010.

G. Restall, “A Note on Naive Set Theory in LP”, Notre Dame Journal of Formal Logic, 1992, 33 (3): 422-432.

R.T. Brady, The non-triviality of dialectical set theory, in G. Priest, R. Routley, J. Norman (Eds.), Paraconsistent Logic, Philosophia Verlag, Munich (1989), pp. 437-471.

Thierry Libert, Models for a paraconsistent set theory, Journal of Applied Logic, Volume 3, 2005, pp. 15-41.

N.C.A. da Costa, O.A.S. Bueno, Paraconsistency: towards a tentative interpretation, Theoria, 16 (40) (2001), pp. 119-145.

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I've always understood this as the essence of Naive Set Theory - Comprehension is essential to the language-first kind of set theory used in schools and common discourse, and we have to either adjust the logic or restrict the language to work around antinomies.

I'm not sure if this is the canonical understanding, but I've definitely seen it at work in e.g. Brady (1983): http://projecteuclid.org/euclid.ndjfl/1093870447

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  • I know about naive set theory, but the article talked about an axiomatic system, like ZFC of NFU. I've upvoted your answer and edited the question accordingly.
    – user42828
    Commented Feb 5, 2022 at 15:29
  • It took me a long time to grasp the significance of differentiating between the language and the theory. I even remember the Eureka moment when I realized the significance (it happened during a discussion on the old Usenet). I don't know how much of this is due to the way set theory is taught and how much is due to the fact that I was a programming languages researcher. I suspect they both had a part.
    – David Gudeman
    Commented Feb 5, 2022 at 23:00
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    (A silly quibble, in case people want to google: the spelling is "antinomy"... :)
    – paul garrett
    Commented Feb 6, 2022 at 20:45

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