Re: Cantor
- From: "Randy Poe" <poespam-trap@xxxxxxxxx>
- Date: 5 Oct 2005 14:00:07 -0700
Top posting repaired.
Pedro wrote:
> "Randy Poe" <poespam-trap@xxxxxxxxx> wrote in message
> news:1128519621.564953.271540@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> >
> > Pedro wrote:
> >> I think I'm getting sidetracked by giving poor examples. Can't see the
> >> forest for the trees, and all that... Let me ask it one more way, if I
> >> may.
> >>
> >> It's obviously impossible to actually present a "complete" list of reals
> >> prior to initiating the diagonalization method as the list is infinitely
> >> long. Rather, the diagonalization method is a thought experiment that
> >> says,
> >> in effect, "no matter how complete one makes the list, one can use this
> >> typographical method to find with certainty a number not on the list".
> >>
> >> But I can make the same conceptual argument about the natural numbers (or
> >> non-negative integers, if you prefer). In my example below, SSSS0 differs
> >> in
> >> the first position from 0, differs in the second position from S0,
> >> etc...,
> >> thereby giving a number that differs from all others on the list and
> >> demonstrating an absurdity: that the natural numbers are not enumerable.
> >> (The only new assumption / constraint I've placed on this example is that
> >> the list be ordered. That doesn't seem to be an unreasonable thing to do
> >> with the natural numbers.)
> >>
> >> So I'm obviously missing something fundamental about the whole
> >> diagonalization thing. Any idea what it is?
> >
> > This point is made below by James Dolan. I'm just putting in
> > my two cents worth.
> >
> > In the "diagonalization" proof that [0,1] is uncountable, the
> > form of the proof is:
> >
> > 1. Let f:N->[0,1] be any mapping from N to [0,1].
> > 2. There exists x in [0,1] such that x does not equal
> > f(n) for any n in N.
> >
> > The construction of x in step 2 relies on a theorem that
> > any string of digits 0.ssssss... represents an x in [0,1].
> >
> > Remember, the purpose of the construction is not to produce
> > a sequence, but to generate a single value x which, by the
> > construction, is not mapped by any f(n). This number x has
> > the property that its n-th digit differs from the n-th digit
> > of f(n).
> >
> > So the corresponding proof in the natural numbers should be
> > trying to construct a corresponding m in N. But you do not
> > have the corresponding theorem that any (possibly infinite)
> > sequence of digits represents a natural number. Therefore
> > the construction can not be guaranteed to produce such a
> > number.
> >
> Thanks. I think I've got it and I appreciate your help and your patience.
>
> In laymans terms:
> - In the case of enumerable things (like natural numbers), I'll eventually
> find it on the infinite list of naturals if only I look long enough. The
> fallacy was in stopping and excalming that because I've not found it yet, it
> must not be there.
I'd say the fallacy was that you were assuming you could
construct an arbitrary infinite digit sequence that would correspond
to a natural number, but perhaps I misunderstood your original
question.
> - In the case real numbers, no matter how big I make the list or how long I
> look there will always be at least one real that can't possibly be on the
> list. And since there are an infinite number of ways to construct that
> missing real, there are an infinite number of reals that aren't on even an
> infinite list.
Right. There are in fact an uncountable number of reals missing
from any countable list.
- Randy
.
- References:
- Cantor
- From: Pedro
- Re: Cantor
- From: Randy Poe
- Re: Cantor
- From: Pedro
- Cantor
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