Undecidability (was reasoning)

[ER <ression@hotmail.com>]

--- In HPFGU-OTChatter at yahoogroups.com, "Tim Regan <timregan at m...>" 
<timregan at m...> wrote:

> --- "ER" wrote:
> can one introduce a new, self-evident, axiom to ZF that allows CH 
> to be decidable? 

Well, I can partly answer my own question here. Apparently, in 2000, 
H. Woodin added a new "plausible" axiom to ZF that allowed the 
Continuum Hypothesis to be shown false. And it seems that set-
theorists feel that this should be the case, which leaves Hermione 
and the Trolley Witch embroiled - eek! I wonder if Woodin knows what 
he has done? Doesn't mathematics stink.

>My understanding of Gödel's Theorem is that your job would never be 
>finished. For any new theorem, you could extend the axioms to make 
>the theorem provable (the easy way is to add the theorem as an axiom 
>itself, but there may be ways that meet your "self-evident" criteria 
>too). But lurking around the corner is another theorem that you 
>cannot prove so you add axioms to fix that. But lurking around the 
>corner is another theorem that you cannot prove so you add axioms to 
>fix that. But lurking around the corner is another theorem that you 
>cannot prove so you add axioms to fix that. If I were Lemony Snicket 
>I'd keep this going for at least a page, but I'm not. Gödel proved 
>that you must either go on forever adding axioms or add one that 
>contradicts an existing axiom rendering the system inconsistent.

Okay, I didn't know that. Sort of answers my question really, thank-
you. Although I do know (or rather, I read) that Godel proved that 
one of the undecidable things in an axiom system is the consistency 
of the axioms! So whether you'd always know you contradicted another 
axiom is a moot point. I think. Lemony Snicket is not a mathematician 
I take it? Too good a name I suppose ... the Lemony Snicket 
Postulate? The Lemony Snicket Lemma, nah, they'd never wear that ;)

>It is odd how Gödel's Theorem hasn't had the impact you'd expect in 
>mathematics, outside the study of logic and foundations. 

Well, I suppose he torpedoed part of Hilbert's wish-list. And if he 
and Cohen closed down the chance of deciding the Continuum Hypothesis 
one way or the other, then you could say that he prevented 
mathematicians from knowing how many real numbers there are!

ER