TITLE: Math without infinity
QUESTION [34 upvotes]: Does math require a concept of infinity?
For instance if I wanted to take the limit of $f(x)$ as $x \rightarrow \infty$, I could use the substitution $x=1/y$ and take the limit as $y\rightarrow 0^+$.
Is there a statement that can be stated without the use of any concept of infinity but which unavoidably requires it to be proved?

REPLY [29 votes]: Surprisingly, infinity proves necessary even for finite combinatorial mathematics.  For a nice explanation as to why there cannot be any such as 
thing as a comprehensive, self-contained discipline of finite 
combinatorial mathematics see Stephen G. Simpson's writeup of his expository talk Unprovable Theorems and Fast-Growing Functions, Contemporary Math. 65 1987, 359-394. 
Simpson gives a detailed discussion of three theorems about finite objects whose
proofs necessarily require the use of infinite sets. The three theorems discussed are about colorings of finite sets (modified finite Ramsey theorem),
embeddings of finite trees (Friedman's finite form of Kruskal's theorem) 
 and iterated exponential notation for integers (Goodstein's theorem).
Below is an excerpt from the introduction.

$\quad$The purpose of the talk is to exposit some recent results (1977 and 
  later) in which mathematical logic has impinged upon finite combinatorics. 
  Like most good research in mathematical logic, the results which I am 
  going to discuss had their origin in philosophical problems concerning the 
  foundations of mathematics.  Specifically, the results discussed here were 
  inspired by the following philosophical question.  Could there be such a 
  thing as a comprehensive, self-contained discipline of finite 
  combinatorial mathematics?
  $\quad$    It is well known that a great deal of reasoning about finite 
  combinatorial structures can be carried out in a self-contained finitary 
  way, i.e. with no reference whatsoever to infinite sets or structures.  I 
  have in mind whole branches of mathematics such as finite graph theory, 
  finite lattice theory, finite geometries, block designs, large parts of 
  finite group theory (excluding character theory, in which use is made of 
  the field of complex numbers), and large parts of number theory (including 
  the elementary parts but excluding analytical techniques such as contour 
  integrals).  One could easily imagine comprehensive textbooks of these 
  subjects in which infinite sets are never mentioned, even tangentially. 
  All of the reasoning in such textbooks would be concerned exclusively with 
  finite sets and structures.
  $\quad$  Consequently, there is a strong naive impression that the answer to 
  our above-mentioned philosophical question is "yes."
  $\quad$ However, naive impressions can be misleading.  I am going to discuss 
  three recent results from mathematical logic which point to an answer of 
  "no."  Namely, I shall present three examples of combinatorial theorems 
  which are finitistic in their statements but not in their proofs.  Each of 
  the three theorems is simple and elegant and refers only to finite 
  structures.  Each of the three theorems has a simple and elegant proof. 
  The only trouble is that each of the proofs uses an infinite set at some 
  crucial point.  Moreover, deep logical investigations have shown that the 
  infinite sets are in fact indispensable.  Any proof of one of these finite 
  combinatorial theorems must involve a detour through the infinite.  Thus, 
  in a strong relative sense, the three theorems are "unprovable" -- they 
  cannot be proved by means of the finite combinatorial considerations in 
  terms of which they are stated.