TITLE: If all sets were finite, how could the real numbers be defined?
QUESTION [41 upvotes]: 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 the 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?

REPLY [4 votes]: Finitism still allows you to use infinitary definitions of real numbers, because a finitist is content with finite proofs even if the concepts mentioned by those proofs would seem to require infinite sets. For example, a finitist would still recognize that "ZFC proves that every bounded nonempty set of reals has a least upper bound" even if the finitist does not accept that infinite sets exist. 
Proofs in various infinitary systems are of interest to finitists because of conservation results. In this setting, a conservation result would show that if a sentence about the natural numbers of a certain form is provable in some infinitary system, the sentence is actually provable in a finitistic system.  For example, there are finitistic proofs that if any $\Pi^0_2$ sentence about the natural numbers is provable in the infinitary system $\text{WKL}_0$ of second order arithmetic, that sentence is also provable in the finitistic system $\text{PRA}$ of primitive-recursive arithmetic. 
Many consistency results are proven finitistically. For example, there is a finitistic proof that if ZF set theory without the axiom of choice is consistent, then ZFC set theory with the axiom of choice is also consistent. This proof studies infinitary systems of set theory, but the objects actually handled are finite formal proofs rather than infinite sets.