# How would calculus be possible in a finitist axiom system?

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
@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