X-Message-Number: 23856
From: "michaelprice" <>
References: <>
Subject: Undeciability is significant
Date: Sun, 11 Apr 2004 11:11:05 +0100

Robert Ettinger wrote:

> Thomas Donaldson wrote in part:
> The Cretan problem ["paradox" of The Liar or Epimenides] ultimately 
> led to the conclusion that there could be unprovable math theorems. 
> Ostensibly. It is true that Goedel likened his undecidability theorem to 
> The Liar, but that was inaccurate, because provability and truth are 
> completely different. 
> Furthermore, Goedel's conclusion was a mere language trick and of 
> no mathematical significance. He only showed that it is possible to 
> label certain sentences in such a way that they are undecidable. 

No, Godel's theorems are of great mathematical significance.
The Generalised Continuum Hypothesis (that there is no transfinite 
between the number of natural numbers and the number of points 
on a line) is one such example of an undecidable proposition.  It 
and its negation(s) are compatible with the rest of set theory, as is 
also the axiom of choice.

Proving the Continuum Hypothesis was problem number 1 on
David Hilbert's famous list of oustanding problems, given at the
International Congress of Mathematicans in 1900, and Hilbert
assumed it could be proved.  He was wrong.  This is a greatly
significant result.

> In a somewhat similar vein, Cantor's definition of "set" allowed 
> nonsense sets.

which lead to the modern axiomatic approach.

Michael C Price

Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=23856