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/nla

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 https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem [1] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf [2] https://arxiv.org/abs/math/0305282 https://arxiv.org/abs/math/0305282