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Re: Peer-reviewed arguments against Cantor Diagonalization
Posted:
Oct 28, 2012 11:27 PM
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On Oct 29, 1:14 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> On Sunday, October 28, 2012 10:01:31 PM UTC-5, Hercules ofZeus wrote:
> > On Oct 29, 12:20 pm, Arturo Magidin <magi...@member.ams.org> wrote:
>
> > > On Sunday, October 28, 2012 9:04:41 PM UTC-5, JRStern wrote:
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> > > > On Sun, 28 Oct 2012 17:41:59 -0700 (PDT), Arturo Magidin
>
> > > > <magi...@member.ams.org> wrote:
>
> > > > >Huh?
>
> > > > >Sorry, but that statement makes absolute no sense
>
> > > > >whatsoever to me. What exactly was the point you
>
> > > > >were attempting to make?
>
> > > > That I am not arguing with the conclusion that Cantor's Theorem is
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> > > > true, I am questioning whether the diagonal argument is coherent.
>
> > > > How can this be unclear?
>
> > > Because I did not simply state the conclusion. I gave you the "diagonal argument". If you are questioning whether it is "coherent", then you should point to whatever point you find incoherent, rather than simply quote and then give a sentence fragment.
>
> > > The government doesn't like it when I read minds without a warrant, so I try not to do it, you see.
>
> > > I have you a complete proof of Cantor's Theorem; the diagonal argument is embedded in that theorem. What is it that you find incoherent?
>
> > > If there is nothing you find incoherent, then why is it that you continue to "question" its coherence?
>
> > > If you could not even tell that you were presented with the argument in the first place, then perhaps your problems arise much sooner than at Cantor's diagonal argument?
>
> > > If it is a *particular* presentation of the argument that concerns you, then it is incumbent upon you to specify which presentation it is you find yourself having doubts about, and stop nattering about "peer-review", books, and the like.
>
> > > --
>
> > > Arturo Magidin
>
> > Are you saying you just proved:
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> > ALL(f):N->R E(r):R ALL(n):N f(n)=/=r
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> > in 1ST ORDER LOGIC?
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> > i.e. FOL = Quantifiers Over Arguments Not Functions.
>
> In ZF, functions are sets; and the objects of the theory are sets. So the statement above is a perfectly fine first order statement in the language of ZF. Moreover, there is a *set* of functions from N to R, so I'm quantifying over the **objects** that are elements of that set.
>
> But surely, if you know about first order logic, then you knew that?
>
> And if you didn't... well, you are someone else whose problems come from much earlier than any objection you might by trying to raise here; your problem is the same as Sir Richard Phillips's.
>
> --
> Arturo Magidin
>
>
And you see no problem with using a logic composed of predicate
strings (functions) to construct sets to construct functions to range
over ALL functions, and calling it 1st order logic - No Quantifying
over functions here!
Herc
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