Re: J class theory with comprehension corrected.

On Nov 22, 2:37 pm, G. Frege <nomail@invalid> wrote:
On Thu, 22 Nov 2007 11:31:18 -0800 (PST), Zaljo...@xxxxxxxxx wrote:

Now why don't you just stick with Ackermann's original Theory?
(Actually, I had the impression that your J class theory was
just a "muddled" version of Ackermann's theory.)

No F. You didn't understand J class theory. J class theory is a trial
to join Mk and Ackermann's, however it failed [...] since it yields an
inconsistent theory.

Let me clarify my statement: I didn't want to claim that J class theory
was just a variant of Ackermann's theory. What I wanted to claim was
that you theory looked like a "muddled" -and hence possibly defective-
version of Ackermann's theory.

So I still would like to suggest that you just stick with Ackermann's
original theory (if you are not able to come up with a own theory which
actually _is_ an improvement compared with Ackermann's version).

F.

Ok, no problem F. , take a theory which have all Ackermann's theory
( with or without Regularity) and add the following axiom to it.

Axiom: Ax Ay ((x e V & ~ y e V) -> x subnumerous_to y).

were subnumerous_to have the standard definition.

Now Can you tell me what is the Cardinality of @ were @ is the class
of all ordinals that are sets, which is of course a proper class.

Is such theory consistent?

The problem with this theory is that whatever class of inaccessible
ordinals you define using 'ZF+inaccessible cardinals' set theory, then
it will be a set in this theory!

So what is the Cardinality of @ were @ is the class of all ordinals
that are sets, which is of course a proper class.

It appears to me that the above additional axiom will inflate V
into a very large cardinal that no set theory defined till now, and
thus V will be the biggest inaccessible cardinal (whose all its
members are sets) present ever.

Can anybody answer this question?

Is there a known cardinal that is larger than V in this theory?

Zuhair








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