×

Numerology in topoi. (English) Zbl 1103.18002

As a footnote on the first page explains, this paper has existed in typescript form since 1981, and the results in it have been known to experts in topos theory for 25 years, but they have only now become available in publicly accessible form. In the interim, some of the author’s ideas concerning his concept of ‘numeral’ have appeared (after independent rediscovery) in “Orbits and monoids in a topos” by J. BĂ©nabou and B. Loiseau [J.Pure Appl.Algebra 92, 29–54 (1994; Zbl 0793.18002)]; there is also an account of them in section D5.5 of the reviewer’s book [P. T. Johnstone, “Sketches of an Elephant. A Topos Theory Compendium. I”, Oxford Logic Guides 43; Oxford Science Publications. Oxford: Clarendon Press (2002; Zbl 1071.18001)]. It is worth mentioning these references, since they give more detailed proofs than Freyd’s paper, which is written in highly telegraphic style, with many proofs omitted altogether. But Freyd’s paper goes further than the references just cited (some of the additional results will be in the yet-to-be-completed third volume of the reviewer’s book), so it is worth making the effort to decipher it. In particular, by studying numerals in the free topos, he proves that the free topos (without natural number object) contains an internal category whose (external) category of global sections is the free topos with natural number object – so that, as Freyd puts it, ‘After one drops excluded middle, there is no great gain in dropping the axiom of infinity’.

MSC:

18B25 Topoi
03F50 Metamathematics of constructive systems
03G30 Categorical logic, topoi

Keywords:

numerals
PDFBibTeX XMLCite
Full Text: EuDML EMIS