# Are there examples of theorems incompatible with ZF that have proven useful in the sciences?

I was thinking about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and I realized all the mathematics I know about in science are compatible with ZF (even if they sometimes take additional axioms). So if I'm correctly understanding the situation the question becomes "why did Zermelo and Fraenkel construct a set of axioms that are useful in the real world", which seems to have a much simpler answer.

However, I'm assuming that ZF is actually what's useful here. However, if there are theorems that are really useful in science but are incompatible with ZF then it starts to look more like mathematics is generally useful.

Also, I get there are other systems that can construct many, maybe all of the useful theorems in science, but that doesn't really change the motivation for my question.

• To modify a Hamming quote, if the structural integrity of a particular airplane depended on a choice of axioms for set theory, I wouldn't fly in it. More seriously, you may find something of interest in the discussion of a similar question at MathOverflow, mathoverflow.net/questions/27428/… – Gerry Myerson Dec 6 '17 at 23:23
• Interesting. I like all this sorts of discussion, thanks for linking that. However, it is a different sort of question from what I was asking. I don't think ZF (with or without the C) exactly describes reality, just is useful for modeling reality. For instance, as far as I know humans have never actually encountered an infinite amount of stuff (countable or not) but the axiom of infinity is certainly required for lots of useful axioms in physics. – zenten Dec 6 '17 at 23:38
• Arg, I meant to say theorems, not axioms. – zenten Dec 6 '17 at 23:46