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Tony Orlow wrote:
> MoeBlee said:
> > Tony Orlow wrote:
> > > Yes, at least standard theory recognizes the power set as a larger beast,
> > > albeit through somewhat sketchy logic.
> >
> > Wnat is sketchy about the logic?
(Why do you write "proof" in quotation marks below? To remind us that
you can't understand it, perhaps?)
> Well, the entire "proof" regarding the power set rests on the notion of the
> element which is not in the set to which it maps, and the contradiction
> derived
> from assuming such a thing indicating that no such element exists in the
> mapping, and that therefore the bijection fails to map an element.
A major difference between cranks and mathematicians with alternative
theories, is that mathematicians can accurately quote elementary
results of theories they are proposing alternatives to. The "proof" as
you laughably refer to it is a proof of the non-existence of a
bijection between two sets. That's all. Mathematicians like proving
results like This exists, or That does not exist, because these results
are simple and incontrovertible, as opposed to the hand-waving we see
in other quarters.
> In the case
> of the power set it's possible to construct such a statement, whereas in other
> situations where a bijection is possible between obviously different levels of
> infinity, such an argument isn't possible to construct. So, it seems there is
> a
> proof that rests one a particular notion, while the general notion of
> quantitative comparisons such as 2^aleph_0 is lost when it comes to 2*aleph_0,
> aleph_0^2, or any other formulaic expression on infinity. So, while I don't
> disagree that the power set is larger, I don't consider the proof to be of the
> sort that leads to very precise comparisons.
What nonsense. Nothing is "lost". Mathematicians are perfectly capable
of formalising other comparisons of infinite sets, and getting it
right.
> It only manages to distinguish
> this particular relative infinity through a peculiarity of the construction.
> Maybe "sketchy" isn't the right word. Maybe the logic is just parochial. I
> don't really disagree with it. I just think we can do better.
Uh, _you_ think _you_ can do better. I think David Ullrich would
probably write "Guffaw" at this point.
Brian Chandler
http://imaginatorium.org
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