# Internal categoricity for schemes

## Mateusz Łełyk

**Authors:** Mateusz Łełyk and Ali Enayat

The talk is devoted to the exposition and explanation of the recent results on the notion of internal categoricity and its various declinations. Within this line of research we aim at understanding how much of the second-order categoricity of various foundational theories (such as Peano Arithmetic, PA or Zermelo-Fraenkel set theory, ZF) can be recovered in the first-order setting? These considerations germinated in the introduction of the concept of internal categoricity of \(\text{PA}\) in full second order logic by Hellman and Parsons, which was followed by Väänänen’s introduction of the notion of internal categoricity of Peano Arithmetic and Zermelo-Fraenkel set theory in the context of Henkin models of second order logic in [6] and in joint work with Wong [5], and later in the context of first order logic, as in [8] and [7]. Internal categoricity has been substantially explored and debated in the philosophical literature, as witnessed by Button and Walsh’s monograph [1], the recent monograph of Maddy and Väänänen [4], and in the recent work of Fischer and Zicchetti [3].

In the talk we argue that the notion of internal categoricity is best seen as a property of schemes, i.e. first-order formulae with a second-order parameter which apply to (first-order) languages returning (first-order) theories. To wit, we prove that each sufficiently strong r.e. theory (in particular: each r.e. extension of PA or ZF) can be axiomatized by a scheme which is *not* internally categorical. Secondly, we introduce a short hierarchy of categoricity-like notions for schemes which refine internal categoricity and are based on Enayat’s notion of solidity ([2]). Using it we prove two general theorems on the preservation of the appropriate properties by adding the full comprehension and the minimality scheme. As a corollary we are able to derive in a uniform fashion the internal categoricity of the \(n\)-th order arithmetic and Burgess’ extension of the untyped Kripke-Feferman truth theory, KF\(\mu\).

## Bibliography

- Button, Tim and Walsh, Sean,
, Oxford University Press, 2018.*Philosophy and model theory* - Enayat, Ali,
*Variations on a {V}isserian theme*,, (J. van Eijk, R. Iemhoff, J. Joosten, editors), Coll. Publ., London, 2016, pp. 99- 110.*A tribute to {A}lbert {V}isser* - , Fischer, Martin and Zicchetti, Matteo,
*Internal categoricity, truth and determinacy*,, vol. 52 (2023), no. 5, pp. 1295- 1325.*Journal of Philosophical Logic* - , Maddy, Penelope and Väänänen, Jouko,
, Cambridge University Press, 2023.*Philosophical Uses of Categoricity Arguments* - Väänänen, Jouko and Wang, Tong,
*Internal categoricity in arithmetic and set theory*,vol. 56 (2015), np. 1, pp. 121- 134.*Notre Dame Journal of Formal Logic* - Vä”{ a}nänen, Jouko
*Second order logic or set theory?*,, vol. 18 (2012), no. 1, pp. 91- 121.*The Bulletin of Symbolic Logic* - , Väänänen, Jouko,
*Tracing internal categoricity*,, vol. 87 (2021), no. 4, pp. 986- -1000.*Theoria. A Swedish Journal of Philosophy* - , Väänänen, Jouko,
*An extension of a theorem of {Z}ermelo*,, vol. 25 (2019), no. 2, pp. 208- 212.*The Bulletin of Symbolic Logic*