Naïve Comprehension in HYPE

(Logic, )

he project of a mathematically tenable naïve set theory is inevitably confronted with the difficulty of finding a suitable nonclassical logic. The proof-theoretic weakness of most naïve set theories and the ad-hocness of some of their tenets, which entail commitment to paracompleteness or paraconsistency, make a theory based on an iterative notion of set look like the best option. I ar-gue that there are some fundamental advantages in adopting a naïve notion of set. A successful naïve set theory must ultimately be based on a framework which allows paradox in a controlled way, retaining classicality in the domain of ordinary mathematics, thus gaining the expressive power given by naïve sets without entirely sacrificing the classical structure and proof-theoretic strength.

A suitable framework is provided by the impossible world approach of the logic HYPE, which allows paracomplete or paraconsistent submodels while maintaining classicality locally in well-behaved situations. Building on recent developments in theories of truth, which show that the theory of truth KFL based on HYPE has the same proof-theoretic strength as its classical counterpart KF, I build an arithmetical theory of naïve comprehension, HYAC. This theory serves as a comparison between theories of truth and theories of naïve comprehension as property instantiation. I high-light the similarities between a notion of set thus construed and a notion of disquotational truth, and show that the theory HYAC based on HYPE has at least the same strength as KFL. I also set to show that HYAC can express compositionality for membership, i.e. iterative or mathematical set formation, hinting that a HYPE naïve set theory built from first principles might achieve a strength similar to that of the iterative set theories. Although the use of the HYPE framework needs to be better justified, the interesting interplay between nonclassicality and strength is worth considering for future developments.

Chair: Yannick Kohl

Time: 10:40-11:10, 11 September 2021 (Saturday)

Location: SR 1.006

Remark: (Online Talk)

Maria Beatrice Buonaguidi

(King_s College London, United Kingdom)

I am currently completing my MSci in Physics and Philosophy at King's College London. I am fascinated by Logic and by the Philosophy of Mathematics in all their aspects, but I am especially passionate about non-classical logics, paradoxes and the interface between logic and metaphysics. I am also inspired by feminist philosophy, and about the issue of representation of women, especially in academia, something which I feel rather personally as an aspiring female logician.

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