Re: Cantor's Donut Paradox

From: Andrew (stan370_at_btinternet.com)
Date: 06/23/04 

Date: Wed, 23 Jun 2004 17:23:19 +0100

"George Greene" <greeneg@greeneg-cs.cs.unc.edu> wrote in message
news:xesbrjc6ai0.fsf@greeneg-cs.cs.unc.edu...
> Will Twentyman <wtwentyman@read.my.sig> writes:
> : The continuum hypothesis can be arbitrarily determined to be true
> : or false. See http://mathworld.wolfram.com/ContinuumHypothesis.html
>
> I have to play dictionary policeman here.
> "Determined" is over-claiming here.
> A better verb would be "stipulated".
>
> : The first line from your abstract is "It is known that if a
> : solution to the continuum hypothesis exists then it lies outside
> : the scope of set theory." This is simply not true.
>
> No, it's true.
>
> : It is stated inside the scope of set theory
>
> Right.
>
> : and "solutions" have already been generated.
>
> No, they haven't. Precisely as you stated at the outset,
> it has been proven that as far as ZF is concerned, there
> simply IS NO "solution". It's consistent either way.
> A solution would constitute DETERMINING (as OPPOSED to just
> stipulating, which is ALL you can do if ALL you have
> is ZF) what "the real" answer "really" is.
> My point being that A Lot Of People out there are of
> the opinion that ZF, like PA, has a STANDARD model,
> has an INTENDED model, that the canonical axiomatization
> (which in ZF's case also includes the axiom of choice)
> only approximates. A *solution* to [the problem of]the
> Continuum Hypothesis would be a "determination" or demonstration
> or discovery of the particular truth-value that the continuum hypothesis
> has in THAT model. In natural language, the intended model of
> ZFC is the one with "full" powersets, the one where
> the powerset of every set Really Does Contain "all" subsets
> of that set.
>
> : Regardless, it lies entirely within the scope of set theory.
>
> No, really, it lies completely outside it, until you come up with
> some more axioms describing set theory's standard model.
> Any particular model in any case DOES BY DEFINITION lie
> outside the scope of the theory.

I determined it! Cantor's continuum hypothesis is screwed in ANY of the
models. The truth value is that the question is illegal, and hence no
solution can become legal, though the specific limited conceptions of the
continuum hypothesis, and its variants, which have been demonstated to be
consistent within set theory are all true, providing they are referenced to
the correct model (even though they may be contradictory taken all
together).

See my site for specifics - www.cantorsdonutparadox.co.uk

>
> : In addition, your Cantor's Cobblers is incorrect. At each stage
> : for the loop, the machine is adding a *finite* number of shoes to
> : the string. It only "reaches" aleph_0 shoes.
>
> It doesn't even reach that far. Unless he is doing something
> very weird with time, his process will simply never finish.
> Even IF it finishes, there is NEVER an aleph_0'th ROUND!
> There is never an individual round in which aleph_0 shoes
> get produced, let alone 2^aleph_0.

I agree with you, but then again I don't. I have also answered this on my
site - including answers to the real world and transfinite arithmetic
arguments.

But to cut it short, see my later post "Aleph nought is finite" for
justification that if Cantor is right, then aleph nought is finite and so
there can be no justification for excluding the aleph nought-th round.

Relevant Pages

  • Re: Cantors Donut Paradox
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    (sci.logic)
  • Re: Aleph One Sets
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