From: Mike Oliver <oliver@math.ucla.edu>
Subject: Re: Continuum Hypotheisis
Date: Thu, 13 Jan 2000 12:18:05 -0800
Newsgroups: sci.math
Summary: [missing]

Daryl McCullough wrote:
> 
> Pardon my ignorance, but is there a simple intuitive
> definition of O#.

Any completely accurate definition is going to be a little
technical.  But in brief:  Suppose there's an uncountable
set of ordinals which are order-indiscernibles for L;
that is, if phi is a formula with n free variables, and
if alpha_0 < alpha_1 <...< alpha_{n-1} and
beta_0 < beta_1 < ... < beta_{n-1} are all taken from
the set of indiscernibles, then

	L |= phi(alpha_0...alpha_{n-1})
		iff
	L |= phi(beta_0...beta_{n-1})

Then 0# is the collection of all codes for
true-in-L formulas of the form phi(alpha_0...alpha_n-1)
where the alphas are indiscernibles.

There's an additional technical property (look up "Silver indiscernibles")
which is needed to make this unique (as stated above, you
might get different 0#'s for different sets of indiscernibles;
but you can't for different sets of Silver indiscernibles).

> I vaguely recall that it has something
> to do with coding a model of set theory as a real, so
> that 0# exists --> ZFC is consistent (but not the other
> way around?)

That's right.