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