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I am interested in learning a little more about finitism, currently about which I only know a few encyclopedic paragraphs.

I know that during some time, some mathematicians like Kronecker thought that finitism is the right choice, so I guess that an important theory such as calculus would somehow be obtained in such an axiom system.

So I have two questions along these lines:

1) Is there a construction of calculus within a finitist axiom system? If so, does it include the important theorems that are taught to a first year student, (like the extreme value theorem, and fundamental theorem of calculus, with an appropriate definition of function)? Are the proofs much more complicated than the standard calculus?

2) Could you give some fundemantal axioms, and define what a function means in such a system? I am especially curious about constructing some real numbers with a definition like this Wikipedia example: http://en.wikipedia.org/wiki/Constructivism_%28mathematics%29#Example_from_real_analysis , but I don't know what a function would mean.

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Well, if you are a sufficiently finite finitist, then you reject calculus itself: math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html –  Qiaochu Yuan Jul 25 '10 at 15:05
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@Qiaochu Yuan: That is not what is usually meant by "finitism", a finitist accepts existence of any natural number. The position that does not accept this is usually called "ultra-finitism". –  Kaveh Jul 25 '10 at 16:53

2 Answers 2

The book Simpson, Stephen G. Subsystems of second order arithmetic. Perspectives in Logic. Cambridge University Press, Cambridge; Association for Symbolic Logic, ISBN: 978-0-521-88439-6 MR2517689 will tell you far more than you want to know about this topic. It explains exactly what assumptions have to be added to a basic finitisitic system to prove various common theorems of calculus. The idea is to start with a basic form of second order arithmetic equivalent in strength to primitive recursive arithmetic (which is what is sometimes meant by finitisitic mathematics) and show that theorems of calculus are equivalent over this weak system to various axioms (such as weak Konig's lemma). You can also check http://en.wikipedia.org/wiki/Reverse_mathematics for some details.

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It is not completely clear for me what is the intended meaning of "a finitist axiom system". AFAIK, Kronecker was not a finitist, but rather a semi-intuitionist. Do you mean something similar to Primitive Recursive Arithmetic (PRA) (which is considered by some experts to correspond to Hilber's finitism?). Do you consider first-order Peano Arithmetic (PA) as a finitist axiom system?

If you mean a system that does not accept existence of infinite objects but only finite numbers/strings/..., then there are various approaches toward mathematical analysis, which would satisfy this condition. For example there is Markov/Russian School of constructivism, there are computability schools, ... . One important school which is completely compatible with classical mathematics is Bishop school, see books by Errett Bishop and Douglas Bridges.

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Thanks for your answer! I am not an expert so I don't know about PRA, but what I mean by a finitist system is exactly as you described in your second paragraph. So I think first order PA is a finitist system. I'm sure your pointers will be very helpful to my question. –  AgCl Jul 25 '10 at 19:08
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The problem is many finitist do not accept quantification over infinite domains, like natural numbers. Since PA includes induction over formulas with arbitrary first-order quantification, they do not accept it as finitist. If you are considering PA as acceptable, then Simpson's book mentioned by Richard is a good reference (though it is a logic book) which tries to find the weakest system containing PRA that one can prove a theorem. PRA is a very similar to PA. Basically, we have function symbols and definitions for all primitive recursive functions, and induction for quantifier-free formulas. –  Kaveh Jul 26 '10 at 11:06
    
By the way, there is also a book by M. Beeson, "Foundations of Constructive Mathematics" which you may also like. It considers different constructive approaches to mathematics. –  Kaveh Jul 26 '10 at 11:15

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