TITLE: Is all ordinary mathematics  contained in high school mathematics?
QUESTION [55 upvotes]: By high school mathematics I mean Elementary Function Arithmetic (EFA), where one is allowed +,×, xy, and a weak form of induction for formulas with bounded quantifiers. This is much weaker than primitive recursive arithmetic, which is in turn much weaker than Peano arithmetic, which is in turn much weaker than ZFC that we normally work in. 
However there seem to be very few theorems (about integers) that are known to require anything more than this incredibly weak system to prove them. The few theorems  that I know need more than this include:
*Consistency results for various stronger systems (following Godel). This includes results such as the Paris Harrington theorem and Goodstein sequences that are cleverly disguised forms of consistency results.
*Some results in Ramsey theory, saying that anything  possible will happen in a sufficiently large set.  Typical examples: Gowers proved a very large lower bound for Szemeredi's lemma showing that it cannot be proved in elementary function arithmetic, and the Robertson-Seymour graph minor theorem is known to require such large functions that it is unprovable in Peano arithmetic. 
I can think of no results at all (about integers) outside these areas (mathematical logic, variations of Ramsey theory) that are known to require anything more than EFA to prove. 
A good rule of thumb is that anything involving unbounded towers of exponentials is probably not provable in EFA, and conversely if there is no function this large then one might suspect the result is provable in EFA. 
So my question is : does anyone know of natural results  in "ordinary" mathematics (number theory, algebraic geometry, Lie groups, operator algebras, differential geometry, combinatorics, etc...) in which functions larger than a finite tower of exponentials occur in a serious way?  In practice this is probably more or less equivalent to asking for theorems about integers unprovable in EFA.
Related links:
http://en.wikipedia.org/wiki/Grand_conjecture about Friedman asking a similar question.
By the way, encoding deep results as Diophantine equations and so on is cheating. And please do not make remarks suggesting that Fermat's last theorem needs inaccessible cardinals unless you understand Wiles's proof.

REPLY [2 votes]: I think we're heading towards an era where "ordinary" mathematics includes logic, at least the parts of it that can be described as applied mathematics (used in practical problems of the real world).  For example, the ML programming language is based on polymorphic lambda calculus.  I'm no expert but I have the impression that the proof that polymorphic lambda calculus is strongly normalizing is equivalent to second order arithmetic.  There are fancier languages based in even more powerful (?) theories, like Coq implements Martin-Lof type theory more or less directly.  I'm just a programmer trying to learn these languages but people disigning them and writing compilers for them (e.g. implementing type inference) seem to me to often be up to their elbows in proof theory. I saw a thesis by someone about ordinal analysis of programs (after all, by the Curry-Howard correspondence, programs are proofs..).  I wondered if software engineers (at least those in high-assurance programming) will someday use proof-theoretic ordinals in their daily work just like electrical engineers now use complex numbers.  
In complexity theory, there are some proofs that P vs NP is independent of some sizeable fragments of PA, but maybe those fragments are weaker than EFA.