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?
Best Answer
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.