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).

Read Dana Scott and Alexander Zenkin's very interesting feedback.

Added Dec. 6, 2008: I am very pleased and honored that Wolfgang Mueckenheim quoted this opinion in lecture XI of his fascinating course.

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 (who teaches at Ramapo College, NJ), who wrote a book called "The Challenge of Infinity". The relevant part of the book is available here.

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