Infinitely many proofs? While compiling a list of my favorite proofs of the infinitude of primes, the following came to mind;
Proposition: There are infinitely many non-isomorphic proofs of the infinitude of primes.
I'm not sure if this is true. Is it? How could one prove (or disprove) this?
I'm worried that because "non-isomorphic" isn't rigorously defined, there isn't much one could say about this. If this is the case, is there any way to clean up the statement to make it amenable to a proof while keeping the same spirit of the proposition?

I think a sensible first step would be to agree on a formal notion of proofs, i.e. on some formally defined system like Natural Deduction. Then you could try to start the proof that there are infinitely many proofs of your proposition on this basis.
If you want to include some notion of non-trivial equivalence of proofs, like  the "isomorphy" mentioned by you, you could establish a rigorous definition based on a formal proof framework. For example, if you have a framework with modus ponens/cut, i.e. the application of lemmas, a natural way (in my opinion) would be to consider those proofs to be in one equivalence class that use the same lemmas.
You could try to formalize the proofs you know of (possibly a tedious job, though) and thereby find the places where the constructions diverge. Maybe there is some point with probably infinitely many possibilities to continue.
