Re: Cantor and the binary tree

In article <MPG.1cfe8e078fe6cf0c989d29@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <1117019416.558035.49530@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >
> > A number
> > > 0.333... is always equal to a rational with a denominator that is a
> > > power of ten. It is simply impossible to assume that this number
> > > becomes 1/3. Or is there any occult advantage of the decimal system
> > > over the binary system?
> >
> > That "number" does not have to BECOME 1/3, it already is 1/3.
> >
> > If one regards 0.333... not as a number but only as a sequence of
> > partial sums, it is true that none of those partial sums is exactly 1/3,
> > but the mathematical standard for interpreting a repeating decimal, like
> > 0.333..., is that it represents that NUMBER which is the limit of that
> > sequence of partial sums. And that number, by every reasonable analysis,
> > is exactly and precisely 1/3.
> >
> > WM must be off his meds again.
> >
> But, Virgil, how do you know that, when you can never get to infinity?

Infinity is not a place, it is meerely a lack of any 'finite' boundary.

> Don't you have to perform all your partial sums?

Not if you can prove a limit exists, or doesn't exist, without doing so.

> Isn't a limit something that never
> gets there?

Wrong way round. The limit doesn't 'get' anywhere.
The sequence of partial sums, if convergent, 'gets' closer to that limit.

> Can you just "jump" to infinity, and declare that infinite set of
> partial sums equal to some fraction?

No! And it is nonsense to speak of an infinite set of values being equal
to some number.

One definition says a number, L, is the limit of an infinite sequence if
and only if all but finitely many values in that sequence are within any
pre-assigned positive distance of L.

More formally: L is the limit of f: N -> R if and only if
for any given positive real epsilon ,
the set {n e N: |f(n) -L| > epsilon} is finite.
.

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