Re: GCH vs. Axiom of Choice.

From: J.E. (troubled6man_at_yahoo.com)
Date: 12/13/04 

Date: 13 Dec 2004 08:36:35 -0800

>Are you assuming that every statement X for which we can prove
>Con(ZF)=>Con(ZF+X) and Con(ZF)=>Con(ZF+~X) is neither true
>nor false? I don't mean it as a rhetorical question.

I think of them as sentance such that (~ZF)v(X) cannot be validities.
As opposed to theorems T of ZF which I consider to be sentances such
that (~ZF)v(T) is a validity.

I define the truth of a sentance based on the truth of atomic
sentances, "z in x" for elements x,z of a domain of discourse, then a
sentance is true or false based on the existence of a winning strategy
for one side in a particular game played where atomic sentances give
winners to the game and choices of elements are from a fixed domain of
discourse. A validity is a sentence that is has a winning strategy
(i.e. is true) in any domain of discourse. If the game is the usual E
and A game, then this a standard view of truth and falseness, if the
game is an IF-logic game with more moves like Ex//y instead of just Ex,
then it's less popular, but not very different IMO.

The concept of a complex sentance being true or false without reference
to a domain of discourse just doesn't make sense to me, it could be
valid (is true in all possible worlds with well defined true/false for
the atomic sentances) inconsistent (is false in all possible worlds
with will defined true/false for atomic sentances) or consistent (true
or false depending on which possible world or equivalently where it's
truth depends).

Examples: Ex Ey x in y is consistent. (Ax Ay ~ x in y )v(Ex Ey x in
y) is a validity for non-empty domains of discourse (and for empty
domains I wouldn't know how to play the game). While Ex Ey ((x in
y)&(~x in y)) is inconsistent for non-empty domains of discourse.

>I don't assume anything like that, and to me this seems like a
>nonsequitur. If you believe such a thing, and you don't know
>why you believe it, it's liable to be a source of some confusion.

I don't even know what it MEANS to say a sentance X is true without
reference to a domain of discourse. Saying it's valid I get. Saying
it's true, I don't get. Saying it's false, I don't get. Saying "for
any nonempty domain of discourse, it is true or it is false" I get and
if the game is that perfect information A and E game, then I even
believe that it's true. That was the basis of boot straping, that I
take sentances like A is a validity where A is formed from composite
sentance B and C, to get binary relations "B in C" that can be
interpreted as "true" when A is a validity, and false otherwise and
thus get a concrete domain of discourse from considering the case of
arbitrary domains of discourse. It's really a kind of
meta-quantification, but if's a possible world that I can understand
that's better than a world that I can't.

So L contains "all the ordinals" that can be proven and any set
generated by a cumulative application of, for instance, separation?
That sounds a bit vague since we need both "all" and "ordinal" before
we could know what was in L.

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