transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
An arithmetic pretopos is a pretopos with a parameterized natural numbers object. A list-arithmetic pretopos is a pretopos with all parameterized list objects (Maietti 10, 2.6).
Using the (equivalent) definition given in Cockett 1990, a parameterized list object is a W-type for the polynomial functor . This definition makes sense since a pretopos has finite products and disjoint coproducts (here denoted “”).
Discussion via its internal language, which is a dependent type theory… (Maietti 05, Maietti 10, p.6).
Maietti (05,10) proposed that list-arithmetic pretoposes serve as the arithmetic universes that André Joyal (cf. Joyal 05) once suggested to use for discussion of incompleteness theorems (cf. van Dijk/Oldenziel 2020); they are used directly as the definition of arithmetic universes e.g. in (Maietti-Vickers 2012).
Robin Cockett, List-arithmetic distributive categories: Locoi, JPAA 66 no.1 (1990) pp.1-29.
Robin Cockett, Finite objects in a locos, JPAA 116 (1997) pp.169-183.
Joost van Dijk, Alexander Gietelink Oldenziel, Gödel’s Incompleteness after Joyal, arXiv:2004.10482 (2020). (abstract)
André Joyal, The Gödel incompleteness theorem, a categorical approach, (abstract) Amiens 2005, Cah. Top. Géom. Diff. Cat. 46 no.3 (2005) p.202. (numdam)
Maria Maietti, Reflection Into Models of Finite Decidable FP-sketches in an Arithmetic Universe, Electronic Notes in Theoretical Computer Science
122 (2005) 105-126 [doi:10.1016/j.entcs.2004.06.054]
Maria Maietti, Modular correspondence between dependent type theories and categories including pretopoi and topoi, Mathematical Structures in Computer Science 15 6 (2005) 1089-1149 [doi:10.1017/S0960129505004962, pdf]
Maria E. Maietti, Joyal’s arithmetic universe as list-arithmetic pretopos, TAC 24 3 (2010) 39-83 [tac:24-03, pdf]
Maria E. Maietti, Steve Vickers, An induction principle for consequence in arithmetic universes, JPAA 216 (2012) pp.2049-2067. [doi:10.1016/j.jpaa.2012.02.040, pdf]
Paul Taylor, Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe, Electronic Notes in Theoretical Computer Science 122 (2005) 247-296 [doi:10.1016/j.entcs.2004.06.059]
Steve Vickers, Sketches for arithmetic universes, arXiv:1608.0159 (2016) &lbrackarXiv:1608.01559]
Steve Vickers, Arithmetic universes and classifying toposes [arXiv:1701.04611]
Last revised on December 13, 2023 at 17:06:44. See the history of this page for a list of all contributions to it.