Is Fermat's last theorem undecidable?
|See Fermat's last theorem and undecidability - useful books for a list of books related to this subject|
I know what you're probably thinking. Either 'Has this guy been asleep for more than ten years and not realised that its been proved?' or more likely 'Is this going to be a crank article which claims that Wiles didn't prove it'. Well you'll just have to read on to find out.
|Fermat's last theorem|
|This is the statement that the equation xn+yn=zn has no integer solution for x,y,z>0,n>2. Fermat claimed to have a proof which the margin was 'too small to contain.' It is likely that he had a proof for the case n=4 and thought incorrectly that a similar proof could be used for all other values of n.|
Cast your mind back...
In the past there was speculation Fermat's last theorem might be undecidable, and it was realised that this would mean that there couldn't be a counterexample (which would be a proof that it was false), and hence that it was true. However, some people wanted it to be really undecidable and so speculated that the question of decidability was itself undecidable, but that also was undecidable, and so on forever. An ingenious idea, with just one slight problem, which was that it was TOTAL NONSENSE.
The Natural Numbers
|A listing of first order axioms for arithmetic can be found in this Wikipedia article|
The behaviour of the set N of natural numbers 1,2,3... and how to perform basic operations on them is one of the first things we learn in mathematics. Thus we get an intuitive knowledge of how they behave. At the end of the 19th century, there was a program to base the whole of mathematics upon collections of axioms. In the case of the natural numbers, these axioms could be considered as 'self-evident truths'. This is different to some other areas of mathematics where a collection of axioms might be proposed just to see what theorems they lead to. Also, in a subject such as set theory, the properties of the system aren't so self-evident. For example we don't have intuitive knowledge of how infinite sets behave.
|Second order axioms|
|Sometimes a number system's axiomatization may include second order axioms, which allow statements about sets of numbers. (e.g. The Real numbers, Peano's original axioms for the natural numbers). This essentially allows some of our intuitive knowledge about the system back in when deciding whether a statement is true or false, and so decidability of these systems is of less interest.|
Godel's incompleteness theorem says that we can never totally encapsulate our understanding of the natural numbers with a collection of (first order) axioms. This means that given such a collection of axioms, it is possible to construct a system obeying those axioms, which is different to the natural numbers, in that some statements are true in the new system which are false for the natural numbers and vice versa. (Such systems are sometimes called supernatural numbers). Although Godel's undecidable statement was an artificial construction (and much to long to be actually written down), in the 1980's some meaningful statements in number theory were shown to be undecidable. These were related to rapidly increasing sequences - in effect the definition of each item in the sequence in terms of the previous ones can be shown to be inexpressible using the axioms of number theory, although we know what it means. In set theory, the continuum hypothesis and the axiom of choice were proved to be undecidable earlier, in the 1960's, but as I said that's somewhat different to number theory. Note that questions of decidability are never themselves proved from a collection of axioms, which is why it was nonsensical to talk of the decidabilty of a statement as being in itself undecidable.
Fermat's last theorem
So could Fermat's last theorem be undecidable from the standard axioms of number theory. Well, firstly this would mean that it was true - which is indeed the case. Secondly that no one would be able to prove this from the standard axioms - which is also the case - the proof requires knowledge of a lot of other areas of mathematics, which in turn rest on our intuitive knowledge of the behaviour of the natural numbers. So it looks entirely possible that it is indeed undecidable. But as for proving it, that's a different matter. The theorem isn't directly linked to a rapidly increasing sequence, but it might be possible to link it to such a sequence .
A New Axiom?
If Fermat's last theorem is undecidable from the current axioms, then the obvious thing to do would be to add more axioms so that it could be proved. But what should we add? Well the simplest thing to do would be to add Fermat's last theorem as an axiom, but that seems to be cheating (it's hardly self-evident). If , however, it could be related to a rapidly increasing sequence, then that sequence would suggest a new axiom which was (reasonably) self-evident, yet independent of the other axioms. This would allow for a proof of the theorem directly from a collection of axioms, and one might hope that such a proof would be simpler to understand than that devised by Wiles. In any case I can't help feeling that Fermat's last theorem hasn't finished generating interesting mathematics.