Re: Separation,Power and Countability.

On Jun 14, 9:20 am, LauLuna <laureanol...@xxxxxxxx> wrote:
Dear Zuhair,

I've noticed you've already sent two 'last' posts; that's really a
harder paradox than uncountability :-).

As Rupert has said, if you take a constructivist or at least a
definitionist point of view, you may well decide to declare non
existent whatever cannot be defined. This has a rationale, if you
think mathematical objects cannot exist outside our minds, i.e. if you
reject Platonism. But that's a philosophical question.

Most constructivists and definionists reject the unrestricted Powerset
axiom. But if you accept it, you're obliged to admit that P(N) is not
countable. I think Rupert is absolutely right on this. Nevertheless,
I've known people who rejected this as well, arguing that (roughly)
the proof of Cantor's theorem uses a non predicative definition (of
the diagonal set).

You may complain that the axioms of ZF have a philosophy of set theory
incorporated. This is hardly news.

Even so you should distinguish mathematical discussion from discussion
on the foundations of mathematics, though this is sometimes hard to
do.

As I see it, constructivism and definitionism meet a problem in
Richard's paradox, which apparently shows that the set of all
definable reals is uncountable.

The word uncountable set is not acceptable to me,because the language
of any set theory T should be countable, otherwise it will affect the
recurssive ability of that theory and by then it will lead to the
emergence of the shadowy sets i.e the indefinable sets, which is un
acceptable. I personally don't know Richard's paradox, but we can
simply overcome this problem by stating that the class of all
definable reals is countable and not a set, i.e it is a countable
proper class.

I have the following formula in my mind.

AxEPAy(yex<->P(y))

which is a second order language formula.
The unrestriced power set axiom is not acceptable at all, it only
leads us to chase shadows.

Zuhair


Anyway, I hope you will stay here with us. I use to carefully read
your posts and I find them interesting because you seem to be
searching, wondering, discussing etc. on your own about these topics.
Fresh thought is always needed.

Regards

Thank you very much. I appreciate it. I will do so if I have the
time.

Farewell

Zuhair

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