Re: Shift-errors, recursive definitions

On May 4, 9:40 pm, "karl malbrain" <malbr...@xxxxxxxxx> wrote:
"Phil" <toob-head...@xxxxxxxxxxxxx> wrote in message

news:463AD78B.6020907@xxxxxxxxxxxxxxxx



Let me give a simple example, which is apparently known as "Berry's
antinomy," taken from "Set theory, logic and their limitations" by Moshe
Machover.

Here's the version from Chaitin:

Clearly, in both cases the paradox arises in some way from a self-reference,
but outlawing all self-reference would be throwing out the baby with the
bath water. In fact, self-reference will play a fundamental role in the work
of Gvdel, Turing, and my own that I'll describe later. More precisely,
Gvdel's work is related to the liar paradox, and Turing's work is related to
the Russell paradox. My work is related to another paradox that Russell
published, which has come to be called the Berry paradox.

What's the Berry paradox? It's the paradox of the first natural number that
can't be named in less than fourteen words. The problem is that I've just
named this number in thirteen words! (Note that the existence of this number
follows from the fact that only finitely many natural numbers can be named
in less than fourteen words.)

This paradox is named after G.G. Berry, who was a librarian at Oxford
University's Bodleian library (Russell was at Cambridge University), because
Russell stated in a footnote that this paradox had been suggested to him in
a letter from Berry. Well, the Mexican mathematical historian Alejandro
Garciadiego has taken the trouble to find that letter, and it's a rather
different paradox. Berry's letter actually talks about the first ordinal
that can't be named in a finite number of words. According to Cantor's
theory such an ordinal must exist, but we've just named it in a finite
number of words, which is a contradiction

karl m

I don't think you understand Garciadiego at all well. He is saying
that the supposed "paradoxes" are not paradoxes at all--they are
nonsense, they have no logical content. The emphasis has now shifted
from these "paradoxes" to the equal idiotic response--natural
mathematics, which says we have to arbitrarily import into arguments
the idea that mathematics is innate--to see where twentieth-century
researchers expressed it in their theories. On that subject...


Ryskamp, John Henry, "Paradox, Natural Mathematics and Twentieth-
Century Ideas" (April 14, 2006). Available at SSRN: http://ssrn.com/abstract=897085


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