Re: Separation,Power and Countability.

On Jun 20, 2:11 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
On Jun 20, 12:37 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

d cannot be equinumerous with PPw.

Right, and I see that I misspoke. What I meant was actually the
opposite of what I wrote. With the generalized continuum hypothesis, d
is equinumerous with Pw, since d is either equinumerous with Pw or
with PPw, and d is not equinumerous with PPw.

if there is injection from Pw to d, then this means that there is a
bijection from Pw to d.

That's right.

More basically, I don't know what you think is at stake in showing
that d is dominated by Pw.

How can d be dominated by Pw.

Oh boy, I'm really backwards today. What I meant is not 'd is
dominated by Pw' (it is) but rather 'Pw is dominated by d' (is it?
you're asking).

Nevertheless, if d is equinumerous with Pw, then d is dominated by Pw.

d cannot be strictly dominated by Pw

Correct, but if d is equinumerous with Pw, then d is dominated by Pw.
You know that 'dominated' and 'strictly dominated' are not the same,
right?

But I asked for anyone in this forum to SHOW me an injection from Pw
to d. were d is uncountable and indefinable subset of Pw.

See my next remark.

of course I am speaking in a set theory without choice.

Then you're not allowing the generalized continuum hypothesis.

Ok no problem lets assume choice.

Assume choice or assume the generalized continuum hypothesis?

And in either case, you're probably not going to "see" a "shown"
PARTICULAR injection. With the generalized continuum hypothesis, it
follows that an injection does exist, but I don't know how one would
show a PARTICULAR injection, which is made even more remote by the
assumption that d is not definable. With the axiom of choice, I don't
know whether it follows that an injection does exist, but if one does,
it seems remote that one could show a PARTICULAR injection given the
non-constructive nature of choice and, again, given the assumption
that d is not definable.

MoeBlee


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Relevant Pages

  • Re: Separation,Power and Countability.
    ... with PPw, and d is not equinumerous with PPw. ... Assume choice or assume the generalized continuum hypothesis? ... PARTICULAR injection. ... it seems remote that one could show a PARTICULAR injection given the ...
    (sci.math)
  • Re: Separation,Power and Countability.
    ... such a difficulty that makes one wonder weather ... a PARTICULAR injection is showable. ... guess that the generalized continuum hypothesis is not constructive. ... Just an intuition. ...
    (sci.math)
  • Re: Separation,Power and Countability.
    ... know that there is a proof that Pw dominates d. ... continuum hypothesis, d is either equinumerous with Pw or with PPw.) ... an injection. ...
    (sci.math)