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](http://meta.math.stackexchange.com/questions/172/why-did-you-close-my-question-if-all-sets-were-finite), 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?
> 
> 
>
['https://math.stackexchange.com/questions/501', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/145/']
There is a fragment of mathematics that is given by a set of axioms known as the [Peano axioms](http://en.wikipedia.org/wiki/Peano_axioms). Using these rules you can carry out a vast amount of mathematics relating to natural numbers. For example you can prove lots of theorems in number theory using these axioms. The Peano axioms make no reference to sets at all, whether finite or infinite. The only things that exist in this theory are naturals. You can't even form the set of all integers. You can only talk about the naturals themselves. So a vast amount of mathematics would work absolutely fine.

Even though Peano's axioms are about naturals, you can already use them to talk about finite sets. The idea is that any finite set could be encoded as a finite sequence of symbols which in turn could be represented as naturals using [Godel numbering](http://en.wikipedia.org/wiki/G%C3%B6del_numbering). So questions like "is this set a subset of that one?" could be turned into purely arithmetical statements about Godel numbers.

So I'm pretty sure that declaring that there is no infinite set would make little difference to people working within the system defined by Peano's axioms. We'd still have all of the natural numbers to work with, we just wouldn't be able to assemble them into a single entity, the set of all natural numbers.

On the other hand, there are theorems that make essential use of an infinite set. Like [Goodstein's theorem](http://en.wikipedia.org/wiki/Goodstein%27s_theorem). Without infinite sets (or a substitute of some sort) it would be impossible to prove this result.

So the overall result would be, I think, that you could still do lots of mathematics fine. The mathematics you could do wouldn't be all that weird. And you'd simply be depriving yourself of a useful proof technique.

By the way, you'd still be able to say many things about real numbers. A real number can be thought of as a Cauchy sequence. A [Cauchy sequence](http://en.wikipedia.org/wiki/Cauchy_sequence) is a certain type of sequence of rational numbers. So many statements about real numbers, when unpacked, are really statements about rational, and hence naturals, but in disguise.

Update: Uncovering precisely what parts of mathematics you need in order to prove things is a field known as [reverse mathematics](http://en.wikipedia.org/wiki/Reverse_mathematics). Hilbert, and others mathematicians, were interested in trying to prove as much mathematics as possible using finite methods. Although it was ultimately shown that you can't carry out all mathematics using finite methods, it's surprising how much you can. [Here](http://www.andrew.cmu.edu/user/avigad/Papers/elementary.pdf)'s a paper that talks about a system called EA which has no infinite sets. Amazingly we can use results from [analytic number theory](http://en.wikipedia.org/wiki/Analytic_number_theory) in EA. This is because propositions about analytic functions can be interpreted as statements about natural numbers.
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.