I recall from my reading of the popular book, Gödel, Escher, Bach some years ago that Hofstadter speculated on the value of Gödel numbers of undecidable theorems. If I recall correctly, he suggested that defining their reciprocals might have meaning with respect to infinitesimals.
Is there any use in this thinking, or will it only lead to heartbreak?
Well, if we're looking at an undecidable statement, then (by definition of "undecidable") it has no proof (and is therefore not a theorem; "undecidable theorem" is an oxymoron). In particular there is no Gödel number of such a proof.
However, in the particular case of a Gödel sentence, which asserts its own unprovability, since the sentence is undecidable, it will be consistent to assume it is false, and thus your original theory extended with the claim "$G$ has a proof" will be consistent and so have a model. This model just won't be the actual integers.
A model for the extended theory will contain something which the model thinks is the Gödel number for a proof of $G$ -- but actually it won't even be a number. The model will, however (unless the original theory was extremely weak), contain the ordinary integers (or something that behaves enough like them as to make no difference), and it will think that its proof of $G$ is greater than each of those ordinary integers.
So in that sense the model contains something that is larger than each of the actual integers, and if our original theory was something that can speak about rational numbers, it will also contain a reciprocal of that (fake) proof. This reciprocal will then lie somewhere between 0 and every actual positive rational, and thus it can be said to be infinitesimal.
(That's a quite roundabout way of procuring an "infinitesimal" element, though; there are much more direct ways of making a non-standard element than to appeal to the incompleteness theorems, and some of those alternatives behave much nicer).