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I have recently had a conversation with a person who considered intuitionism to be a valid alternative for the "usual" kind of mathematics. Clearly, intuitionism differs from the type of mathematical thinking that I (and probably most of us) are used to, at the very fundamental point of rejecting the axiom of choice.

What puzzles me, is how essential this difference is. I don't quite know the right language, so let me try and explain by example. Consider the Axiom of Choice (C): It is known that C is independent of the Zermelo-Fraenkel axioms, in the sense that if ZF is consistent, then also C is consistent. It follows that if something is true in ZFC, then it may not be provable in ZF, but surely it is not false. Even more, if I somehow knew that a statement cannot possibly depend on C (say, it is very "finite" in nature), then one might as well try and prove it in ZFC rather than ZF (or in ZF + $\neg$ C, for that matter), and the net result would be the same. I would like to know if something similar holds for relation between intuitionistic thinking and the ordinary mathematical thinking. In particular: could it possibly be the case that the typical logic is inconsistent, while the intuitionistic one is sound? Is there a way to somehow "model" the ordinary mathematics within intuitionistic mathematics?

(It is not the main part of the question, and it's terribly open-ended, but if it is the case that intuitionism could be "valid" is some essential case, while the ordinary mathematics could be invalid, I welcome comments on how seriously people take this possibility. I am asking because I always assumed that intuitionists were essentially a fringe group among the mathematicians, and apparently I was not quite correct.)

Jakub Konieczny
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    This is far from my area so I'm not really qualified to give an answer, but one can model different logical systems in maths within the standard model using topos theory. A quick google found some notes here, I don't know how useful they'll be. – tharris Apr 18 '13 at 13:56
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    There is a lot more to intuitionism than unhappiness with the axiom of choice. Rejection of using the excluded middle in arguments is in my opinion much more central. – Marc van Leeuwen Apr 18 '13 at 14:25
  • Pretty much. Diaconescu's theorem forces the intuitionist to reject (full) choice because it implies LEM. – nomen Nov 06 '13 at 09:03

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There are many flavor of intuitionism, and the answer is more complex than it seems. In the context of the natural numbers, intuitionism and classical mathematics have a lot in common. It is only when other infinite sets such as the real numbers are considered that intuitionism starts to differ more dramatically from classical mathematics, and from most other forms of constructivism as well. But basically, there are intuitionistic results that cannot be proved in standard logic and there are standard logic results that are not accepted by intuitionists. This a quick and short answer, an expert on the subject will probably give you a more detailed answer soon. Meanwhile, you can find a nice article here. Wikipedia gives a fine review too.

Wolphram jonny
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  • Thank you for the answer. There is a point in what you write that puzzles me: You wrote that "there are intuitionistic results that cannot be proved in standard logic". Do I understand correctly that you need to add some extra axioms/rules for that? Surely, deleting the rule of excluded middle won't allow you to prove more than previously? – Jakub Konieczny Apr 18 '13 at 14:06
  • You are correct than in most cases intuitionism is more limited because it does not accept many of results of standard math. However, it establishes the existence of the continuum, a continuum having properties not shared by its classical counterpart. I am not an expert, but I guess you could change (rather than add) some of the axioms of set theory so that both concepts become the same. It not that intuitionism proves "more", but rather that it defines some concepts in a different, not mutually interpretable way. – Wolphram jonny Apr 18 '13 at 14:17
  • For related discussion about intuitionistic results that don't have classical counterparts, see http://math.stackexchange.com/questions/176279/all-real-functions-are-continuous – Peter Smith Apr 18 '13 at 15:31