Monday, January 27, 2014
Fermat's Last Theorem and Large Cardinals. Really!
A brilliant math ugrad at UMCP, Doug, is also a creative writer who
wants to work on large cardinals. His creative writing may help him there.
We had the following conversation:
DOUG: The proof of Fermat's last theorem depends on the existence
of certain large cardinals and hence is not in ZFC.
BILL: That is not true.
DOUG: Have you read the proof?
BILL: No, however, if that were true I would know it. See this blog entry.
DOUG: Why would you know it?
BILL: If FLT required LCs then
a) Number theorists would be nervous.
b) Logicians would be ecstatic
c) The math community would not have announced to the world that FLT was solved.
d) Wiles would not have collected his prize money for solving it.
e) Again, I would know it.
DOUG: All compelling arguments. Even so, FLT requires LCs.
BILL: I will bet you five dollars that the current proof of FLT does not depend on LCs.
DOUG: Uh. Your counter arguments are compelling.
BILL: So... no bet?
DOUG: Uh. No.
The next day I got an email from Doug with the subject heading
I cheated myself out of five dollars.
Doug found this article, What does it take to prove Fermat's Last Theorem? Grothendieck and the logic of number theory by Colin McLarty, from 2009.The article says that YES the current proof of FLT DOES depend on LCs. Note that the proof is quite long and uses lots of other stuff that is sort of buried in it. So--- whats the catch?Why aren't number theorists nervous and logicians ecstatic? According to the article anyone who reads the proof of FLT and wanted to could unwind it and get it down to ZFC (and likely down to PA). But nobody has bothered yet.Hence nobody is nervous or ecstatic.
I will take their word for it, but it does make ME nervous. NOT about FLT which I am sure enough people have looked at (and looked at the background literature) that it really can be made to work in ZFC.I am more worried about papers that are not quite so looked at as having LC assumptions that are hidden from the reader that cannot be easily removed.
However KUDOS to Doug for telling me something in math that I did not know and should have. I will treat him to a more-than-five-dollar-lunch.
Postscript: AH, the article was right: FLT was proven using ordinary set theory last year. (See here) by Colin McLarty (I assume its the same person). I will still take Doug to lunch- in a stupid, pedantic, technical sense I was right- the current (2014) proof of FLT did not use LC. But for the real issue of there being any problem at all, I was clearly wrong. Only a logician would say I was right. Hmm- Doug is a logician. Its up to him. (Hmm- my spell checker allows `Hmm' but not `Hmmm')