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.
Best Answer
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.