and my former student Toby Bartels (who just got his PhD) is one. Most such people don't believe in the law of excluded middle, so ZF is right out. And, I believe most of them don't believe you can well-order uncountable sets, because I've never heard of any way to "construct" a well-ordered uncountable set, in the technical sense of "construct".
I can also imagine various sorts of reasonable finitists:
meaning people who don't believe in unbelievably large finite numbers. Unfortunately, it seems hard to develop good axioms formalizing this view, perhaps because the normal concept of proof allows arbitrarily long proofs. I know Christer Hennix and Alexander Esenin-Volpin have tried, but I don't know how far they've gotten. Edward Nelson hasn't worked much on ultrafinitism, but he has expressed sympathetic views in his book "Predicative Arithmetic". In his article "Mathematics and Faith":
I must relate how I lost my faith in Pythagorean numbers. One morning at the 1976 Summer Meeting of the American Mathematical Society in Toronto, I woke early. As I lay meditating about numbers, I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers, leaving me like an infant in a crib reduced to counting on my fingers. Now I believe in a world where there are no numbers save that human beings on occasion construct.
Personally I don't advocate any of these positions, and like Tom Leinster I am happy that you can do mathematics without "believing in" any specific axiom system.
Edward Nelson is a mathematical physicist at Princeton who like me was a student of Irving Segal. I never discussed logic with him, though he read and critiqued my senior thesis when I was an undergrad, and this thesis was on applications of recursive function theory to quantum mechanics.
I used to argue heatedly with Christer Hennix, because he regarded all mathematics using infinity as a sham. I should have spent my time asking him how Esenin-Volpin's alternative system was supposed to work. But our discussions weren't a total waste, because I met my wife through a friend of his - Henry Flynt:
known as the inventor of "concept art", musician, and cognitive nihilist. I'm not sure Henry Flynt would want to be characterized as a "reasonable person".
I only met Esenin-Volpin a couple of times. Besides being the son of the famous Russian poet Sergey Yesenin and the main proponent of ultra-intuitionism, he is known for being a topologist, a dissident during the Soviet era, and a political prisoner who spent a total of 14 years in jail and was exiled to Kazakhstan for 5. His imprisonments were supposedly for psychiatric reasons, but Vladimir Bukovsky has been quoted as saying that Volpin's diagnosis was "pathological honesty":