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Topic: Is continuum completely filled up?
Replies: 610   Last Post: Jun 9, 2012 10:57 AM

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 Randy Poe Posts: 5,658 Registered: 12/6/04
Re: Is continuum completely filled up?
Posted: Feb 28, 2007 2:59 PM

On Feb 28, 2:16 pm, "toshiaki" <fara...@gmail.com> wrote:
> "Randy Poe" <poespam-t...@yahoo.com> wrote in message
>
>

> > On Feb 24, 12:15 pm, "toshiaki" <fara...@gmail.com> wrote:
> > > "Randy Poe" <poespam-t...@yahoo.com> wrote in message
>
>
> > > > > > Let's go one step at a time. Define S_n := sum_{j=1}^n 9/(10^j).
> What
> > > is
> > > > > > the limit of the sequence {S_n} according to the above definition?
>
> > > > > The above definition says that a sequence {a_n} has the limit L if
> for
> > > every
> > > > > eps > 0, there is a natural number M such that, for all natural
> numbers
> > > n,
> > > > > if n > M, then |a_n - L| < eps.
>
> > > > That's right.
>
> > > > > This doesn't mean that for some number n, a_n = L.
>
> > > > That's right. There is no requirement that the limit be a
> > > > member of the sequence. In general, the limit is not. It
> > > > is a number associated with the sequence. It is a fixed
> > > > value. But it is not an element of the sequence.

>
> > > This means that the sequence is not infinite.
>
> > What?
>
> > Did you understand what I said? A sequence is a set of numbers
> > in a particular order. An infinite sequence need not contain
> > every possible number. In particular, an infinite sequence
> > does not need to contain its limit.

>
> > How the HECK does that imply the sequence isn't infinite?
>
> > a. Here is an infinite sequence: 1, 1/2, 1/4, 1/8, 1/16, ...
>
> > b. All of the members of this sequence are of the form
> > 1/2^n where n is an integer >=0.

>
> > c. This sequence is infinite.
>
> > d. The limit of this sequence is 0.
>
> > e. 0 is not a member of this sequence. There is no n such
> > that 1/2^n is 0.

>
> Then this sequence is not infinite.

This sequence is infinite. There is no last value of n.

> > Now, why do you say that d and e imply c is false?
>
> We can conclude from e that all terms of sequence are finite and have
> its successors.

Yes.

> You may imagine infinite space. If you don't get to the
> end of the space, you are at finite position forever.

You confirm what I suspect: That even though we have defined
the actual meaning of "limit", you keep substituting your private
definition as something like a point that the sequence
actually achieves.

That is not the meaning of "limit". Once again, a limit
L of a sequence {S_n} is a value such that for any
epsilon>0, there exists M such that |S_n - L | < epsilon for
all n > M.

the members of the sequence are clustering. They
may or may not ever take that value. All that is
required is that if you wait long enough, all the
sequence values will be as close as you like to that
value.

The meaning of statement "d" is "let epsilon be
10^-20. Then eventually all the values of the sequence
are less than 10^-20. Let epsilon be 10^-100. Eventually
all the values of the sequence are less than 10^-100.
Let epsilon be anything. Eventually all the values of
the sequence are less than epsilon."

Nowhere in the definition of "limit" is there a
requirement that the sequence achieve that value.

stem from substituting your definition of "limit" for
the actual definition.

Can we agree on the actual definition of "limit"?

- Randy

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