From: bevans@carina.unm.edu (Mathemagician)
Subject: Re: Undecidability and the FAQ
Date: Sat, 05 Sep 92 13:26:55 GMT
Organization: Society for the Preservation of E. coli

In article <TORKEL.92Sep4132357@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <hjmn=cl@lynx.unm.edu> bevans@carina.unm.edu (Mathemagician) writes:
>   >Thus, it does provide a counter-example to the FAQ which states that
>   >the undecidability of a statement is, in itself, undecidable.

>  You're going in circles. Permit me to quote an earlier posting of
>yours, in reply to my pointing out that what the FAQ entry states
>is that it is undecidable "in the formal system" whether an undecidable
>statement is undecidable:

Correct.  I am stating that such a statement is incorrect.  Given an
axiom system assumed to be consistent, there are statements that can
be shown to be undecidable.

>   >Incorrect.  It is quite decidable "in the formal system" whether an
>   >undecidable statement A is undecidable.

>  So let me repeat once more. It is not provable in ZFC that the
>continuum hypothesis is undecidable in ZFC. This is a consequence of
>Godel's second incompleteness theorem. Assuming, of course, the
>consistency of ZFC.

Godel's second theorem of incompleteness is that the statement "is
this axiom system consistent?" is undecidable.

From _Mathematics:  The New Golden Age_

What Godel proved was that, for any consistent axiom system which is
strong enough to allow for the development of elementary integer
arithmetic, there will always be statements relevant to that axiom
system which can be neither proved nor disproved from those axioms
(First Incompleteness Theorem).  Moreover, amongst those unprovable
(from the axioms) statements is the statement that the axiom system is
consistent - so the all-important (for Hilberts' programme) notion of
consistency is destined to remain for ever elusive (Second
Incompleteness Theorem).

>   I await with interest any further attempt on your side to avoid grasping
>this simple point.

I await with interest why you claim that CH isn't known to be
undecidable if we assume ZF is consistent.  The point is quite simple
to grasp:  we cannot prove from the axioms that the axioms are
consistent.

To which I say, so?  We must assume they are for an inconsistent
system is of little use.  Again, *of*course* we assume that the axiom
system is consistent.  If it isn't, we cannot be certain of the
answers we arrive at.

Therefore, assuming that ZF is correct, the CH is undecidable.  You
said that, yourself.

How, then, can you say that the undecidability of CH is undecidable
from ZF?  You are moving up one rung on the "meta-ladder."  Yes, we
cannot know if ZF is consistent by staying within ZF.  So?

Tell me, directly...don't just say "Second Incompleteness Theorem" and
expect that to be enough because obviously you aren't communicating
your ideas to me...what it is that is wrong in saying that, given ZF,
the CH is undecidable.  Claiming that it is unknown, within ZF,
whether or not ZF is consistent doesn't make a difference...we're
assuming they are consistent since an inconsistent system is of little
use.

>   Your remaining comments are irrelevant to the point at issue.

I disagree.

-- 
Brian Evans                |     "Bad mood, bad mood...Sure I'm in a bad mood!
bevans@carina.unm.edu      |      I haven't had sex...*EVER!*" -- Virgin Mary