[dan-robertson]

I’ve heard rumours of a course of five or six lectures Conway did many years ago (before he went to Princeton). The goal is to prove Gödel’s incompleteness theorem but somehow using geometry in some way. I think you need to allow comparing two lines and knowing which is longer (and this foundational thing is hard or maybe impossible) in some sense. I don’t know what is left of eg Gödel numbering or the formal languages stuff because I’ve only heard vague descriptions of this.

I believe the contents of this course were lost to time but I’d like to be surprised.

[lidHanteyk]

It isn't what you're looking for, but perhaps Lawvere's generalization of Gödel's work is sufficiently geometric for you [0][1][2].

[0] https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem

[1] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf

[2] https://arxiv.org/abs/math/0305282

[spindle]

I saw Conway talk on Gödel’s incompleteness theorems shortly before he went to Princeton, and the proof he gave was the standard one (although very entertainingly and efficiently presented).