Re: induction vs Cantor

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 11/26/04 

Date: Fri, 26 Nov 2004 15:41:02 -0700

In article <_gMpd.1532$XQ2.39@twister.rdc-kc.rr.com>,
 "Poker Joker" <Poker@wi.rr.com> wrote:

> "Virgil" <ITSnetNOTcom#virgil@COMCAST.com> wrote in message
> news:ITSnetNOTcom%23virgil-4DC867.12412326112004@[63.218.45.211]...
> > In article <8vHpd.1507$XQ2.823@twister.rdc-kc.rr.com>,
> > "Poker Joker" <Poker@wi.rr.com> wrote:
> >
> >> Let L_1 be a list of reals that implies a mapping F_1
> >> between the naturals and reals.
> >
> > For purposes of dicussing Cantor, may one asume that the mapping is a
> > listing of reals, i.e., a function from the naturals to the reals?
>
> In actuality, a listing is infinitely many relations between the naturals
> and
> the reals.

Then you definition of "listing" does not match Cantor's definition, and
youor analysis of your listings is irrelevant to Cantor's construction..
>
> >>
> >> Let D_n be a Cantor anti-diagonal number that can be
> >> formed using the mapping F_n
> >>
> >> Let L_n+1 be a list of reals by inserting D_n into L_n
> >> at row 2n and shifting down all the previous rows at 2n
> >> and above. This process is clearly an inductive process
> >> that creates a new mapping for each natural number.
> >
> > The point of the Cantor "anti-diagonal" construction is that the same
> > simple construction rule simultaneously defines for EVERY list of reals,
> > a real number not listed in that list.
> >
> > So it is not just a particular listing that fails to be surjective but
> > ALL listings simultaneously that fail to be surjective.
>
> That may be so (Simultaneously? What do you mean by that?
> Individually they all certainly fail. They also individually fail at
> every given moment in time if that's what you mean), but my point
> is that there are only countably many anti-diagonal numbers to
> prove failure. So taken together, they form a countable set.
> That set, along with the original list, form a countable set.

Your assumption that there are only finitely many anti-diagonals begs
the very question that we are dealing with. That would seem to require
that the number of functions from the integers to the reals be
countable, but since the set of such functions, R^N, must have greater
cardinality than N, you cannot assume that result.

You need specific proof that the set of ALL diagonals, not just those
you produce by some sequence of operations, is countable, and you have
not done that.

>
> > Let N be the set of naturals and R be the set of reals, and R^N be the
> > set of all functions from N to R, then
> > Cantor:R^N -> R is a function from R^N to R such that
> > for each f in R^N, Cantor(f) is not a member of Range(f).
>
> Are you saying that the "anit-diagonal" numbers don't form a
> countable set? It appears we are avoiding that question from
> the previous paragraph. I see no reason to move on before we
> get that settled.

I am saying that there is no reason to suppose that the set of all
anti-diagonals need be countable. It might be so, I suppose, but it
cannot be assumed to be so.
>
> > This is very similar to the proof that there is no injection from any
> > set to its power set, q.v.
>
> If you mean Cantor's diagonal proof is similar, yes it is. I think
> it has a similar conclusion

Since one can show that the set of reals is of the same cardinality as
the power set of the integers, one cannot show that the set of reals has
the same cardinality as the set of integers given that pwoper sets are
always of greater cardinality than their source sets.

Relevant Pages

  • Re: induction vs Cantor
    ... a listing is infinitely many relations between the naturals ... >> simple construction rule simultaneously defines for EVERY list of reals, ... >> ALL listings simultaneously that fail to be surjective. ... Since one can show that the set of reals is of the same cardinality as ...
    (sci.logic)
  • Re: Cantors "proof"
    ... I gave a listing of all the reals just to try and help out Keckman. ... bijection between the integers, or some subset of the integers, and the ...
    (sci.math)
  • Re: induction vs Cantor
    ... >> between the naturals and reals. ... a listing is infinitely many relations between the naturals ... That set, along with the original list, form a countable set. ...
    (sci.math)
  • Re: induction vs Cantor
    ... >> between the naturals and reals. ... a listing is infinitely many relations between the naturals ... That set, along with the original list, form a countable set. ...
    (sci.logic)
  • Re: concurrency driving me crazy
    ... > Boudewijn Dijkstra wrote: ... >> A listing says more than a thousand words. ... messages will still be queued somewhere and the assertion will fail. ... Prev by Date: ...
    (comp.lang.java.programmer)
Quantcast