Written: Nov. 23, 2005.
David Hilbert famously said:
"No one shall expel us from the paradise that Cantor has created for us."
Don't worry, dear David and dear Georg, I am not trying to kick you out. But, it won't be quite as much fun, since you won't have the pleasure of my company. I am leaving on my own volition.
For many years I was sitting on the fence. I knew that it was a paradise of fools, but so what? We humans are silly creatures, and it does not harm anyone if we make believe that א zero, א one , etc. have independent existence. Granted, some of the greatest minds, like Gödel, were fanatical platonists and believed that infinite sets existed independently of us. But if one uses the name-dropping rhetorics, then one would have to accept the veracity of Astrology and Alchemy, on the grounds that Newton and Kepler endorsed them. An equally great set theorist, Paul Cohen, knew that it was only a game with axioms. In other words, Cohen is a sincere formalist, while Hilbert was just using formalism as a rhetoric sword against intuitionism, and deep in his heart he genuinely believed that Paradise was real.
My mind was made up about a month ago, during a wonderful talk (in the INTEGERS 2005 conference in honor of Ron Graham's 70th birthday) by MIT (undergrad!) Jacob Fox (whom I am sure you would have a chance to hear about in years to come), that meta-proved that the answer to an extremely concrete question about coloring the points in the plane, has two completely different answers (I think it was 3 and 4) depending on the axiom system for Set Theory one uses. What is the right answer?, 3 or 4? Neither, of course! The question was meaningless to begin with, since it talked about the infinite plane, and infinite is just as fictional (in fact, much more so) than white unicorns. Many times, it works out, and one gets seemingly reasonable answers, but Jacob Fox's example shows that these are flukes.
It is true that the Hilbert-Cantor Paradise was a practical necessity for many years, since humans did not have computers to help them, hence lots of combinatorics was out of reach, and so they had to cheat and use abstract nonsense, that Paul Gordan rightly criticized as theology. But, hooray!, now we have computers and combinatorics has advanced so much. There are lots of challenging finitary problems that are just as much fun (and to my eyes, much more fun!) to keep us busy.
Now, don't worry all you infinitarians out there! You are welcome to stay in your Paradise of fools. Also, lots of what you do is interesting, because if you cut-the-semantics-nonsense, then you have beautiful combinatorial structures, like John Conway's surreal numbers that can "handle" "infinite" ordinals (and much more beyond). But as Conway showed so well (literally!) it is "only" a (finite!) game.
While you are welcome to stay in your Cantorian Paradise, you may want to consider switching to my kind of Paradise, that of finite combinatorics. No offense, but most of the infinitarian lore is sooo boring and the Bourbakian abstract nonsense leaves you with such a bitter taste that it feels more like Hell.
But, if you decide to stick with Cantor and Hilbert, I will still talk to you. After all, eating meat is even more ridiculous than believing in the (actual) infinity, yet I still talk to carnivores, (and even am married to one).
Added March 27, 2011: Read Wolfgang Mueckenheim's fascinating book ! I especially like the bottom of page 112 and the top of page 113, that prove, once and for all, that (at least) the actual infinity is pure nonsense.
Clarification added Aug. 25, 2011: My endorsement of Wolfgang Mueckenheim's wonderful book is purely philosophical. I have no expertise, or interest, in checking any possible technical claims that he may have made.
Added Dec. 9, 2019: Another infinity debunker is Professor Eli Rapaport (firstname.lastname@example.org, who teaches at Ramapo College, NJ), who wrote a book called "The Challenge of Infinity". He will be glad to email it to you.
Doron Zeilberger's Homepage