Re: Separation,Power and Countability.

On Jun 18, 11:33 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On Jun 18, 10:03 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Jun 18, 7:19 am, zuhair <zaljo...@xxxxxxxxx> wrote:

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

There is no bijection from w to Pw\{0}. You need a proof of that?
There is no bijection from w to Pw, but Pw is equinumerous with Pw
\{0}, so there is no bijection from w to Pw\{0}.

That's an indirect proof. I am not talking about that.
I am talking about a diagonal formula of Pw\{0}.
what you presented here has nothing to do with what I am asking.





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

Show me EXACTLY what single theory you think proves that there exists
an x such that both x is countable and x is not countable.

MoeBlee- Hide quoted text -

- Show quoted text -

I see it interesting to know that most of the members of Pw are
indefinable, and what I mean by indefinable is
any set x for which the following property hold:
~EPAy(yex<->P), P here is called the defining formula.
I will call such sets 'indefinable sets' or simply
'indefinables'.

So these indefinables in Pw are all subsets of w of course, but these
subsets doesn't have any formula that can define them. I wonder how
the equinumerousity of these subsets with w is proved? It is clear
that all these indefinables in Pw are infinite subsets of w, but how
can we prove that they are equinumerous to w.
Lets say that d is an indefinable subset of w.
Now we have Ef(f:d->w,f is injective), simply because
f(x)=x would do the job.
But how do we prove the opposite direction i.e.
Eg(g:w->d,g is injective).
The problem is that d is indefinable, so there is no rule by which we
can have such an injection, perhpas I am confused here, since I think
that the existence of an injection from w to any subset of w depends
on definability of that subset using separation,i.e
if d is a subset of w,then we can find an injection from w to d if and
only if d is definable . I think I should be wrong. Since it appears
somewhat clear that d even if it is indefinable still it appears that
there should exist an injection from w to d, and by then d should be
equinumerous to w.

So my question specifically is the following:
Question: Suppose d is an indefinable subset of w.
is there a proof of Ef(f:w->d,f is injective)?


Zuhair

.

Relevant Pages

  • Re: Separation,Power and Countability.
    ... definability, but rather we talk about definability in a meta-theory, ... about the set theory in question. ... So these indefinables in Pw are all subsets of w of course, ... can have such an injection, perhpas I am confused here, since I think ...
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  • Re: Simple Set Theory question
    ... finite subsets of x is the power set of x, ... Okay, I see how I can formalize that. ... I pretty much see intutitively that it is an injection, ... equinumerousity by Shroeder-Bernstein), use a Goedel ...
    (sci.math)