Some more comment on Godel's theorem: In my Estonia notes on Turing categories there is a section in which I prove Godel's theorem for a Turing category with a "provability" predicate. http://pages.cpsc.ucalgary.ca/~robin/talks/estonia-winter-2010/estonia-notes... It also contains some general comments about the Godel's theorem (which I hope you like :-)). See section 7.5. The main point of the theorem from a categorical perspective is to show that one cannot have a provability predicate (extensional) and a setting in which there are only two subobjects of 1 (namely empty subobject and the whole thing). This is dramatic enough as it means mere truth and falsity does not suffice there must be stuff in between (lots of it)! The theorem in the notes is a bit more general than Joyal's theorem but also weaker in the sense that there is an assumption of a "provability" predicate. Well you can't have your cake and eat it! Hope this helps ... -robin (Robin Cockett) ________________________________ From: Steve Vickers <s.j.vickers@cs.bham.ac.uk> Sent: Monday, January 16, 2017 3:33 AM To: Patrik Eklund Cc: Categories; joyal.andre@uqam.ca Subject: categories: Re: Theorem or Paradox Dear Patrik, There is a categorical account of Goedel's theorem, by Joyal, and dating back the 1980s (?). I first saw it presented by Gavin Wraith in 1985. The Goedel gap between truth and provability is presented as an issue of internalizability. The logic adequate for expressing arithmetic is obviously not ordinary finitary logic, which cannot characterize the natural numbers. Instead it is identified in categorical terms with "arithmetic universes". Categorically they have nnos and support free algebra constructions. But that is enough to show that an arithmetic universe has, internally, its own initial arithmetic universe. By nesting this construction, considering the initial arithmetic universe in the initial arithmetic universe, one can make the comparison between truth and provability and construct a Goedel sentence. At least, that's my understanding of it. As far as I know the work is still unpublished, and in the outline I have seen there are steps that I believe but don't know how to prove, even though I am actively working on arithmetic universes. I'm not the fastest of mathematicians. It's perhaps also worth noting that arithmetic universes support coherent logic, not full Boolean logic - no negation or implication. Anyway, what I'm saying is that if you want to see category theory evaluate Goedel, then you should probably start with Joyal's work. All the best, Steve. On 13 Jan 2017, at 07:22, Patrik Eklund <peklund@cs.umu.se> wrote:
... I believe we [= The Category Theory Community) can settle this thesis by means of mathematics and category theory, ...
As you may have seen, I tend to believe that the thesis [G??del is wrong, so Hilbert's question remains open] is _right_, ...
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