Among the most fascinating results of mathematics are unprovability theorems, that is, rigorous proofs that certain statements cannot be deduced from certain axioms. A very famous example is Paul Cohen's demonstration that the continuum hypothesis cannot be deduced from the ZFC axioms . For this, Cohen invented a technique known as forcing, which is far too advanced for a page like this. (Indeed, I am incapable of presenting it anyway - if you are curious you could try visiting this site for some notes on forcing. They seem all right, but I don't know enough to be able to judge with any confidence.) Instead, I shall present here a few examples of low-level unprovability theorems, by which I mean purely mathematical results that, in one way or another, tell us that proofs of certain theorems must necessarily have certain properties. Such conclusions I shall loosely refer to as metamathematics.
The looseness is necessary, because any attempt to pin it down seems to lead to a definition that is hopelessly wide. For example, whenever one proves a theorem under certain assumptions, and finds a counterexample to the same statement with the assumptions relaxed, one has shown that the proof of the theorem in some sense `had to use' the assumptions. However, this common situation usually feels like straight mathematics, with nothing particularly `meta-' about it.
Perhaps what singles out the examples I shall look at is that, although most of them can be regarded as merely constructing a counterexample to some purely mathematical statement, it is not initially obvious how to formulate this statement in order to answer some previously arising metamathematical question such as, `Did my proof really have to be this complicated?'
Here, then are the examples I have thought of so far. The first four are very standard and would appear in most undergraduate mathematics courses. The fifth, though not standard, is easy to follow. The others are more advanced, and for them I shall give general descriptions rather than full details - my aim is just to convince you that it is possible to do and understand interesting metamathematics without being a set theorist.
Only the first three examples are written up so far - for the others I give a very brief preview of what I will write when I get the time.