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/']
Set theory with all sets finite has been studied, is a familiar theory in disguise, and is enough for most/all concrete real analysis.

Specifically, Zermelo-Fraenkel set theory with the Axiom of Infinity replaced by its negation (informally, "there is no infinite set") is equivalent to first-order Peano Arithmetic. Call this system *finite ZF*, the theory of hereditarily finite sets. Then under the Goedel arithmetic encoding of finite sets, Peano Arithmetic can prove all the theorems of Finite ZF, and under any of the standard constructions of integers from finite sets, Finite ZF proves all the theorems of Peano Arithmetic. 

The implication is that theorems unprovable in PA involve intrinsically infinitary reasoning. Notably, finite ZF was used as an equivalent of PA in the Paris-Harrington paper "A Mathematical Incompleteness in Peano Arithmetic" which proved that their modification of the finite Ramsey theorem can't be proved in PA.

Real numbers and infinite sequences are not directly objects of the finite ZF universe, but there is a clear sense in which real (and complex, and functional) analysis can be performed in finite ZF or in PA. One can make statements about $\pi$ or any other explicitly defined real number, as theorems about a specific sequence of rational approximations ($\forall n P(n)$) and these can be formulated and proved using a theory of finite sets. PA can perform very complicated induction proofs, i.e., transfinite induction below $\epsilon\_0$. In practice this means any concrete real number calculation in ordinary mathematics. For the example of the prime number theorem, using complex analysis and the Riemann zeta function, see Gaisi Takeuti's *Two Applications of Logic to Mathematics*. More discussion of this in a MO thread and my posting there:

<https://mathoverflow.net/questions/31846/is-the-riemann-hypothesis-equivalent-to-a-pi-1-sentence>

<https://mathoverflow.net/questions/31846/is-the-riemann-hypothesis-equivalent-to-a-pi-1-sentence/31942#31942>

Proof theory in general and reverse mathematics in particular contain analyses of the logical strength of various theorems in mathematics (when made suitably concrete as statements about sequences of integers), and from this point of view PA, and its avatar finite set theory, are very powerful systems.
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.