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Tony Orlow wrote:
> Randy Poe wrote:
> > Poker Joker wrote:
> >> "Dik T. Winter" <Dik.Winter@xxxxxx> wrote in message
> >> news:J6CsBJ.Jys@xxxxxxxxx
> >>> In article <070Tg.14143$8_5.3402@xxxxxxxxxxxxxxxxxxxxx> "Poker Joker"
> >>> <Poker@xxxxxxxxx> writes:
> >>>> "Randy Poe" <poespam-trap@xxxxxxxxx> wrote in message
> >>>> news:1159494111.724651.95600@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> >>>>
> >>>>> That's incorrect. You don't have to assume none map onto R in order to
> >>>>> prove none map onto R.
> >>>>>
> >>>>> The direct argument starts this way: Let f be any such function, from
> >>>>> naturals to reals.
> >>>> Certainly we should assume that f *MIGHT* have R as its image, right?
> >>> You may assume that, but that assumption is not needed.
> >> Certainly not for ostriches.
> >>
> >>>>> Now, are you saying that somehow that misses some possible functions
> >>>>> from naturals to reals? How so?
> >>>> No, but we haven't proven that the image of f can't be R in step #1,
> >>>> right?
> >>>> So step #2 isn't valid, right?
> >>> Remember:
> >>>> 1. Assume there is a list containing all the reals.
> >>>> 2. Show that a real can be defined/constructed from that list.
> >>>> 3. Show why the real from step 2 is not on the list.
> >>>> 4. Conclude that the premise is wrong because of the contradiction.
> >>> Why is step 2 invalid?
> >> Do you always accept steps that have questionable validity?
> >
> > Why does step 2 have "questionable validity"?
> >
> >>>> Under the most general assumption, we can't count out that
> >>>> R is f's image, so defining a real in terms of the image of
> >>>> f *MIGHT* be self-referential, and it certainly is if the image
> >>>> of f is R.
> >>> What is the problem here?
> >> I assume you accept this proof that there are no complete lists
> >> of reals:
> >>
> >> Let r be a real number between 0 and 1. Let r_n denote the nth digit
> >> in r's decimal expansion. Let r_n = 5 if r_n = 4, otherwise let r_n = 4.
> >
> > That doesn't make sense. You are saying that every digit of r
> > both is equal to 4 and is equal to 5.
> >
> > Consider r = 0.00000000...
> >
> > So you're saying the first digit of r is 4 because the first digit of
> > r isn't 4? What the hell are you talking about?
>
> Duh. Sounds like PJ's constructing an anti-diagonal.
>
> >
> >> r isn't on any list of reals. Therefore there isn't a complete list of
> >> reals.
> >
> > That bears no resemblance at all to a proof.
> >
> > - Randy
> >
>
> It bears much resemblance to Cantor's second regarding uncountability
> of...a set. The original proof was regarding a complete language using
> at least two symbols, m and w, no?
Why don't you sketch this resemblance in detail. In particular,
the part that says that any symbol that equals m equals w, and
any symbol that equals w equals m, OK?
- Randy
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