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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 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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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 all 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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