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 As far as I know the work is still unpublished, and in the outline I have seen there are steps that I believe but don't know how to prove, even though I am actively working on arithmetic universes.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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