Re: abundance of irrationals!)

Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message news:<ITSnetNOTcom#virgil-DA055E.13091204042005@xxxxxxxxxxxxxxxxxxxxxxxx>...

> > > If you define distinct objects to be 'a' set, you are in trouble. Do you
> > > mean to define them as separate sets?
> >
> > I meant that 1, e.g., can be defined to be "a set" {{}}.
>
> So is that a yes or a no?

A Yes. 1 is a set, 2 is a set, 3 is a set, ...

> > I did not speak of the limit for n --> oo but asked you whether there
> > is a natural number n which satisfies the equation 1/n = 0.
>
> In fact you asked two questions, which do not have the same answer.
>
> You asked if the sequence contained its limit, to which the answer is no,
> and you asked whether that limit was 0, to which you incorrectly said
> "Certainly not" and which I correctly answered "Certainly so"!

So there was a misunderstanding. But now all is clear and ok.

> > is 1/9. Does 1/9 appear in the diagonal whereas 1/9 does not appear in
> > any line.
>
> That looks like at least two questions to me.

In fact it is the same, because every series can be undersood as a
seqence.
>
> If, by "diagonal", you mean the non-terminating presumably decimal
> expression formed by taking then nth digit from the nth member of the
> sequence as its nth digit, ad infinitum, then that is the decimal
> expansion of 1/9.

In case of series, the limit is contained? Whereas in case of
sequences the limit is not contained (in general)? Ad infinitum is
meant in the sense that only fnite natural numbers appear as indices.
oo does not!
>
> This is merely another example of the limiting value of a sequence not
> appearing as a member of the sequence. But we already had an example of
> that above.

Why, then, do you claim that the diagonal of my list is 1/9? Please
explain.
>
> The fact that the limit of a sequence need not be a member of the
> sequence has been long known, and is not a problem to mathematicians,
> however much it may be to such as you.

Why do such as you claim that the sequence of partial sums does
include 1/9?

Regards, WM
.

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