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June 9[edit]

Categorical Peano axioms[edit]

I'm interested in Peano axioms#Interpretation in category theory but there are no sources listed. Does anyone know where I can find a more complete exposition? --RDBury (talk) 12:17, 9 June 2019 (UTC)[reply]

Maybe here. See also the links/xrefs from that article. 173.228.123.207 (talk) 19:19, 9 June 2019 (UTC)[reply]

I've since found Natural number object as well. The link is interesting and the nLab site in general is worth bookmarking, but all the talk about topoi and sheafs (sheaves?) is a bit out of my comfort zone. I guess what I'm specifically looking for is a proof, not necessarily referencing category theory, that axioms 7-9 in the Formulation section are equivalent to something like:

10: Given a set X, p∈X and a map f:X→X, here is a unique map r:N→X such that r(0)=p and r(S(n)) = f(r(n)) for all n.

In category jargon 10 says (N, 0, S) is an initial object in the category of Pointed unary systems (if I'm using the jargon correctly). My understanding is you need set theory to prove the 7-9⇒10 direction even if you assume X=N; nothing too fancy but it's not trivial. Not sure about the 7-9⇐10 direction. --RDBury (talk) 02:20, 10 June 2019 (UTC)[reply]