From: bevans@carina.unm.edu (Mathemagician)
Subject: Re: Undecidability and the FAQ
Date: Fri, 04 Sep 92 10:34:02 GMT
Organization: Society for the Preservation of E. coli

In article <TORKEL.92Sep1134136@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <zjjnqz_@lynx.unm.edu> bevans@carina.unm.edu (Mathemagician) writes:
>   >I'd like to know why you maintain that the Continuum Hypothesis isn't
>   >known to be undecidable.  Are you claiming that Godel and/or Cohen
>   >used principles outside of ZF?

>   I have maintained nothing of the sort. You're still confused, but
>there is no reason why you should be. Just try to think about
>what you're reading.

I have thought about what I'm reading...the problem is that you don't
seem fit to say everything you mean.

>  Since the statement "CH is undecidable in ZFC" implies that ZFC is
>consistent, it is not provable in ZFC that the continuum hypothesis is
>undecidable in ZFC, by Godel's second theorem.

Didn't you read my statement that of course we are assuming ZFC is
consistent?  I am not implying that ZFC is consistent by saying that
the CH is undecidable.  I'm saying that if we assume ZFC is
consistent, which is the pragmatic thing to do, and is the logical
thing to do since these are the axioms, then we reach the conclusion
that the CH is undecidable.

>   Hence ZFC and the continuum hypothesis afford no counterexample to
>the assertion in the FAQ entry, that it is not provable in the formal
>system itself that an undecidable statement is undecidable.

>   Clear so far?

No, not clear.  I am under the impression that you think I am stating
that I'm going from the results Godel and Cohen achieved to stating
something about the axioms.  However, I am doing the opposite.  I am
starting from the assumption that the axioms are consistent and then
coming to the conclusion that the CH is undecidable.

Thus, it does provide a counter-example to the FAQ which states that
the undecidability of a statement is, in itself, undecidable.

However, the CH is a counter-example.  The fact that we have to assume
the consistency of ZF is not a blow to that since that is the
pragmatic and logical thing to do.  If ZF is inconsistent, how do we
know that the proofs of Godel and/or Cohen are correct?  The system is
inconsistent.

>   What is provable in ZFC, and in fact also provable in elementary
>arithmetic, is the hypothetical statement "if ZFC is consistent, then
>CH is undecidable in ZFC".

Which is what I've been talking about.  As I said, *of*course* we are
assuming that our axioms are consistent.  At the present moment, we
have no reason to believe otherwise.  That doesn't tell us that they
*are* consistent, but we have to work with what we're given.  It could
easily be that ZF is inconsistent which throws into doubt the results
of Godel and Cohen, but we've moved up a step on the "meta" ladder.
As Dr. Devlin goes on to state, Cohen's result showed that just like
there are multiple bases for geometry, there are multiple bases for
set theory.  In some of them, the CH is decidably true.  In some, the
CH is decidably false.  In this particular instance, however, it is
undecidable.  And *all* of these things rest on the assumption that
the axioms are consistent.

An inconsistent set of axioms is of little use.

>   So what "casual formulation" did I have in mind? Simply the casual
>formulation "Godel and Cohen proved that the continuum hypothesis is
>undecidable in ZFC". This is a casual formulation, because Godel and
>Cohen did not actually prove this and never claimed to have done so.
>What they did prove was the hypothetical statement, "if ZF is consistent,
>then the continuum hypothesis is undecidable in ZFC".

Perhaps you call them "casual formations" because you don't understand
all the implications in the paragraphs quoted.  Perhaps I haven't
included enough for you to understand what they are saying.  At any
rate, at no time during this discussion was I straying from the fact
that we are assuming that ZF is consistent.

-- 
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
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