Re: Separation,Power and Countability.

On Jun 17, 10:19 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
On Jun 17, 11:58 am, zuhair <zaljo...@xxxxxxxxx> wrote:





On Jun 17, 10:41 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Jun 17, 7:41 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Jun 17, 6:47 am, LauLuna <laureanol...@xxxxxxxx> wrote:

Eliminate the variable f by
giving it a constant name,

Maybe that's justified in your logic; but it's not justified in mine.

I was wrong; you're right, LauLuna. We can make that move to a
constant symbol (I'm so used to using variables that I forgot, e.g.,
Enderton's version of rule EI). zuhair's revised separation schema
does not block Cantor's argument.

MoeBlee

Yeah, actually I suspected that from the start.
However I still think that separation is the
culprit. I know we can keep separation and revise
power(see the last versions I suggested in this thread)
but still I am not satisfied with this restriction
on power. If we want to add definability axiom
then we should change separation to something else,
to an axiom that doesn't allow Cantor's proof, otherwise
well end up with an inconsistent theory.

By the way is xew&~xef(x) a stratified formula?
were f in injection from w to P(w).

I still think that modifying separation is essential for definability.

Zuhair

Separation can be modified such as to block Cantor's argument using
different ways, like not allowing negative formulae to be sued in
separation, or not allowing non stratified formulae etc....

Anyhow. I have a question.

Question: Working in Z.
if we call ~xef(x) were f:w->P(w),f is injective, if we call it the
DIAGONAL FORMULA FOR P(w).

To be more clear the diagonal formula of A were A is uncounatble,
refers to the formula in separation that proves the uncountability of
A.

My question is that we know that the diagonal formula for
P(w) is ~xef(x).

Now what is the diagonal formula for P(w)\{0}
Also what is the diagonal formula for P(w)\{w}

certainly it is not ~xef(x).

On the contrary, it is; but f is now an injection from w to P(w)/{0}
or to P(w)/{w}

since this fails when all members in w are in their
images in the case of P(w)\{0}, since d={xew&~xef(x)}
were f:w->P(w)\{0} is injective. then d=0 which is not in
P(w).

0 (the empty set) is certainly an element of P(w) since it is a subset
of w. But there is no bijection f from w to P(w)/{0} such that all
values are contained in their images. The range of f includes the
singletons of all members of w except 0. If you wish any element x of
w to be in f(x) you must pair each x with its singleton, except 0. Now
you are left with just one element of w, namely, 0, to pair with the
rest of P(w)/{0}.

If you want to work inside ZF and modify it in a suitable way, you
must reject Powerset.

You should perhaps investigate non cantorian sets in Quine's NF:
http://plato.stanford.edu/entries/quine-nf/

Regards


.

Relevant Pages

  • Re: Separation,Power and Countability.
    ... constant symbol (I'm so used to using variables that I forgot, e.g., ... zuhair's revised separation schema ... However I still think that separation is the ... If we want to add definability axiom ...
    (sci.math)
  • Re: Separation,Power and Countability.
    ... constant symbol (I'm so used to using variables that I forgot, e.g., ... zuhair's revised separation schema ... However I still think that separation is the ... To be more clear the diagonal formula of A were A is uncounatble, ...
    (sci.math)