X-Message-Number: 10233
Subject: A math education for Bob.
Date: Fri, 14 Aug 1998 11:15:17 -0400
From: "Perry E. Metzger" <>

> From: 
> In some town (not Seville) there is one barber, and the barber shaves all
> those men who don't shave themselves. Who shaves the barber?
> Obviously, no problem arises except in the case of the barber himself, so the
> statement essentially reduces to: "If the barber shaves himself, then he does
> not shave himself; and if he does not shave himself, then he does shave
> himself." This is merely self-contradiction, not paradox in any sense.
> Childishly simple, with no shades of Goedel whatever.  

I'm afraid that is not correct.

First of all, you have to understand the significance of the Barber
paradox. The Barber paradox is the reason that early set theory got
demolished. At one time, it was thought to be reasonable in set theory
to speak of any set as though it could be reasoned about -- one could,
for instance, postulate the "set of all sets". Russell's discovery of
the Barber paradox was significant since it destroyed the notion of
the universal set, and indeed almost any arbitrarily constructed
set. From then on, set theory (especially naive set theory) depended
on rules from which to construct new sets given old ones, and accepted
that not all sets were constructible. This was a major revolution. (If
the set theorists had been thinking, they might have seen the analogy
to the fact that not all strings in any formal system can be derived
from the axioms of the system, but that is leaping ahead decades to
Goedel himself.)

Second of all, you seem to believe that the Barber paradox and
Goedel's string (which is approximated in English as "I am not
provable in system X") are somehow different in that the first is a
"simple contradiction" and the latter is not. However, this is not
true. Both are nearly identical in so far as they have only one
direction in which they can be interpreted. In the Barber paradox, one
gains the knowledge that the original assumption was wrong --
translated into set theory, one may not construct the set of all
self-containing sets (or the Universal set), and the initial statement
at the beginning (that we had such a set) was false. Similarly, one
may readily determine the truth of Goedel's string, because its
disproof is contradicted. If Goedel's string *is* false, then the
formal system of mathematics in which it is demonstrated false itself
is inconsistent. (We know, of course, from Goedel, that we may not
prove any consistent mathematics to be consistent, but that is very
different from proving a mathematics inconsistent. One leaves
uncertainty, the other leaves no doubt!)

> Peter Merel (#10221) points out that questioning Goedel and other
> eminences is presumptuous.

Preposterous is more like it.

> The full technical "reasoned argument" that he reasonably requests
> is not yet in satisfactory form--it's very difficult to make it
> clear enough

If it isn't in a satisfactory form, what makes you think that it is
formally correct?

> Goedel himself said his theorem is analogous to the Liar (Epimenides
> etc.) "paradox,"

Analogous, yes.

> and the latter, while still in dispute, was deflated not only by
> little old me, but by many others long ago, including Aristotle. The
> sentence "This statement is false." is not a proposition, because it
> is essentially meaningless--there is no root referent.

You obviously have no understanding of formal systems, Bob.

Mathematics performed in English is a vague approximation of the
"true" mathematics, which is conducted entirely in formal systems. No
one actually does math that way because it is too tedious, but in
principle you have to be able to reduce all mathematical reasoning
into formal systems in order to show that it is correct.

Formal systems are rules for the mechanical transformations of initial
strings (the "axioms") into new strings, the so-called "theorems" of
the formal system.

Goedel did his work entirely on formal systems.  He showed that,
provided the formal system was sufficiently "powerful" (that is,
contained rules of inference known to be needed for doing any real
mathematics) you could always construct a self referential string in
such a formal system and get it to speak about itself.

In such a system, there is no wiggle room for the sort of logical game
you are trying to play. The rules of inference are mechanical. The
game is not one of playing with English words, but playing with
strings via mechanical methods. Interpreting strings of a formal
system into English is something that one does in the same way that
Muslims claim (speciously, possibly) that translating the Koran is
accomplished -- at best, the translation is an approximation of the
original, and is not the definitive text. The reasoning must be done
with mechanical transformations on the strings, not with sophistry in
the English.

By the way, Goedel understood the problem of getting the self
reference to work very clearly. A good deal of the complexity of the
proof comes from the trick he uses in order to get the formal string
to point back at itself. Hoffstadter playfully characterizes the
process as "Quining", after the great logician Quine.

Goedel's two great syntheses were, of course, in demonstrating that
you could turn all formal systems into numbers so that any formal
system in which you would care to do math could be forced to do
mechanical reasoning about itself, and then to demonstrate that you
could force a string into self reference.

The Liar paradox and Barber paradoxes were just an inspiration to
Goedel. His string, which may be constructed in any sufficiently
powerful formal system, exists. It is not a theorem of any formal
system (assuming the system is consistent) but it is true. You can
pretend that this string doesn't exist, but that doesn't make it go
away. You can try to reason it away with tricks in English, but as I
noted, English isn't the language in which this game is played. (Even
Ancient Greek won't do.) Claiming you don't "believe" it makes no
difference. If you accept formal systems as the underlying basis for
logic and mathematical reasoning, then you have to accept the
existence of Goedel's string, and the implications thereof.

> I know, there is no agreement on what constitutes a "proposition,"
> but surely anyone can see at least the possibility that Aristotle
> and I are right,

No, I'm afraid I can't.

> which opens up at least the possibility that Goedel's conclusion was wrong.

You don't even understand Goedel, from what I can tell.


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