Re: Separation,Power and Countability.

On Jun 18, 8:22 am, LauLuna <laureanol...@xxxxxxxx> wrote:
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.

what is the proof of that. since if I use Cantor's argument ~xef(x)
will yield 0, but 0 is in not in Pw\{0}
so there is not proof that every injection from w to Pw\{0} is not
surjective, using the cantorian method.
I do think that there should be a formula other than
~xef(x) that suit the purpose of proving that every injection from w
to Pw\{0} is not a surjection, and that's what I want to know here.



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.

No, we don't need to do so.

Now
you are left with just one element of w, namely, 0, to pair with the
rest of P(w)/{0}.

so what. that doesn't prove anything.

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

If I don't revise separation, then yes you are right.
But If one succeed to revise separation in such a manner as to block
Cantor's argument but without much debilitating separation,then there
is no need to reject Powerset,since by then Pw will be countable and
pose no problem with definability at all.

Still I believe the main anemy to definability is not
power, it is separation, because take for example the set of all
definable subsets of w, you might think this is countable, but
actually with unrevised separation this set is paradoxical since it
can be proved to be countable and not countable ( see with unrevised
separation still Cantor's argument will hold for the set of all
definable sets of w) that's why I say that the problem with
definability is deeper than rejection of power.

rejection of power is a dead end.

While revising separation to fulfill the aims I suggested above, will
revolutionize everything,since there would be only one infinite
cardinal. The concept of uncountability will disappear. Replacement
would be falsified. and thus the concept of set and proper class also
revised, an axiom such as size limitation in NBG would be paradoxical,
choice will hold for all sets,etc...

Separation should be revised, not power.

Zuhair

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

Regards- Hide quoted text -

- Show quoted text -


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