An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitist system, by easy adaptations of the standard definitions. But in order to do any significant mathematics, we need to have definitions for the irrational numbers that one is likely to encounter in practice, such as e or sqrt(2). In standard constructions, real numbers are defined as Dedekind cuts, or Cauchy sequences, which are actually sets of infinite cardinality, so they are of no use here. My question is, how would a real number like those be defined in a finitist axiom system. (Of course we have no hope to construct the entire set of real numbers, since that set is uncountably infinite.)
After doing a little research I found a constructivist definition in Wikipedia http://en.wikipedia.org/wiki/Constructivism_(mathematics)#Example_from_real_analysis , but we need a finitist definition of a function for this definition to work. (Because in the standard system, a function over the set of natural numbers is actually an infinite set.)
So my question boils down to this: How can we define a function f over the natural numbers in a finitist axiom system?
Original version of this question, which had been closed during private beta, is as follows:
If all sets were finite, how would mathematics be like?
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athough the class of all natural numbers would not be a set. In the same sense, we could have the rational numbers. But could we have the real numbers? Can the standard constructions be adapted to this setting?