TITLE: Does Zorn's Lemma imply a physical prediction?
QUESTION [22 upvotes]: A friend of mine joked that Zorn's lemma must be true because it's used in functional analysis, which gives results about PDEs that are then used to make planes, and the planes fly. I'm not super convinced. Is there a direct line of reasoning from Zorn's Lemma to a physical prediction? I'm thinking something like "Zorn's Lemma implies a theorem that says a certain differential equation has a certain property, and that equation models a phenomena that indeed has that property".

REPLY [4 votes]: Pretty much all mathematics of relevance to physics can apparently be formalised in a strongly finitist foundation. Working in 

... a fragment of quantifier-free primitive recursive arithmetic (PRA) with the accepted functions limited to elementary recursive functions. Elementary recursive functions are the functions constructed from some base arithmetic functions by composition and bounded primitive recursion.

Feng Ye's Strict Finitism and the Logic of Mathematical Applications (free draft copy, see also this review (paywall)) builds enough mathematics to treat pretty much all of applied mathematics/theoretical physics (say, that part of theoretical physics that is experimentally confirmed :-)
So as others have pointed out, logical axioms of the strength of full Zorn's lemma are far from necessary.