Intuitionism and rejection of standard logic postulates

Intuitionism refuses the Cantor hypothesis about continuum as a hypothesis meainingless from the intuitionistic point of view Also, 'the tertium non datur' principle: $$A\vee\neg{A}$$ is rejected 'a priori' in the sense that we can only prove the validity of $A$ or $\neg{A}$. My question is: given the rejection of these principles, in particular the second, is it still possible to keep the validity of the mathematical analysis within the intuitionism? Or maybe, accepting the intuitionistic approach, you need to develop a new kind of analysis, because, for example, it's impossible to use the 'reductio ad absurdum' to prove a theorem? Thanks

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1 Answer

There are indeed formal logical systems for intuitionism, just as there are formal logical systems for classical logic. These systems do have slightly different inference rules than classical logic. In the most common cases, though, a set of rules for intuitionistic logic becomes a complete set of rules for classical logic by simply adding the law of the excluded middle as one more rule. So there are not always large differences between the two systems.

By the way, "real numbers", "set of real numbers", "countable", and "bijection" all make sense as terms to intuitionists, and so the statement of CH as "any set of real numbers that is not countable has a bijection with the set of all real numbers" is not meaningless to them. It might be that the statement is neither proved nor disproved - so that they will not call it true or false - but that does not make it meaningless more than any other open problem.

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I think you have to be careful about the notions of infinity within intuitionism. Certain formulations -- and I believe these include those defended by Brouwer and Weyl -- rejected outright the existence of any "actual infinity". From this standpoint "the set of all natural numbers" is a meaningless concept, let alone "the set of all real numbers." There are forms of intuitionism that allow for the actual existence of infinite collections, but I don't think this is a settled issue. –  Arthur Fischer Feb 29 '12 at 12:47
There are certainly many forms of "intuitionism", and some of them reject any actual infinity. But the systems used by perfectly standard books such as Bishop and Bridges Constructive analysis leave CH as a well-posed (although perhaps not interesting or solvable) question. More recently, there has been much work on constructive set theories such as IZF and CZF, which also allow CH to be directly stated, although perhaps not as an interesting question. It may be that the original question was motivated by descriptions of finitistic intuitionism, but I don't see much of that around today. –  Carl Mummert Feb 29 '12 at 13:55
Today at least, Weyl's predicativism is often taken to be "predicativism given the natural numbers". That is, we begin with the assumption that the set of natural numbers exists, but we proceed predicatively beyond that point, avoiding quantifiers over $P(N)$ but allowing arbitrary quantification over $N$. –  Carl Mummert Feb 29 '12 at 13:56
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