FOM: RE: RE: Proper Names and the Diagonal Proof
montez at rollanet.org
Wed Jun 26 16:21:03 EDT 2002
Dennis Hamilton wrote:
``I had thought he used the proof to show that there are more of the things
he wanted to name than the available supply of names.''
``I don't think duplicates are the issue.''
I once believed the same thing you suggest in your first statement above.
Your second statement above is quite clearly on target, IMO.
In the early 1990s, I had occasion to sling back a few with George McNulty
in Hamilton Ontario. I was arguing vociferously that the reason for
Goedel's Undecidability Results was essentially a lack of enough names for
the subsets of the natural numbers. (I got my PhD in 1989, and I was still
very much a Plebe when I went to Hamilton. This situation has improved some
infinitesimal amount in the meantime.) George tried very hard to convince
me otherwise, and I went away very puzzled by the experience. The beers
helped however, for they loosened something in my brain, I guess. It took
me several years to learn and realize that the very theorem I loved so much
as a graduate student shows that one need not even get involved with
undecidability in particular to see that it is not the lack of names that is
the problem. I cannot recall the details of George McNulty's many
explanations about the reasons for Goedel's undecidability reseults, but a
few catch expresions have stayed with me: He said I should imagine one
set-theoretic universe inside another, and the ``poor schmuck'' down inside
the smaller one cannot ``see'' the whole truth about his or her universe.
In particular, Mr. Schmuck is unable to get hold of a proof that his or her
universe makes sense. Later on, I realized that this is similar to Skolem's
Paradox, and the theorem that elucidates this situation is the
Lowenheim-Skolem Theorem. Basically, in some denumerable set-theoretic
universes, although there are ``enough names'' for the sets of natural
numbers, the ``way to find them'' is somehow hidden from the ``poor
Schmuck'' who resides in the given denumerable universe, while his or her
parent, in Cantor's Paradise, can see their entire universe for what it is -
a miniature model of reality constructed from the sticks and stones that
comprise the arsenal of the eternal pedanticist.
The fact that McNulty did not convince me on the spot suggests to me that I
was at the time such a pedanticist. But, I must have been struck in a
beneficial way by George's multitude of comments about the situation, for I
have come to learn some bit from it, and I have come to realize there was a
connection between all these facts.
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