Re: infinity


Tony Orlow (aeo6) wrote:
> Randy Poe said:
> >
> > Tony Orlow (aeo6) wrote:
> > > Taking the sum of all finite naturals depends on identifying the last of them
> > > and completing the sum, but we all know it is impossible to identify any
> > > largest finite.
> >
> > OK, let me try to avoid any discussion of "largest finite"
> > altogether while trying to pin down what properties you
> > believe the set of finite natural numbers has.
> >
> > I'm also going to avoid summation notation.
> >
> > 1. Let S be the set of finite natural numbers, expressed in
> > base 10. Is S a finite set?
> If you are limiting the decimal digit strings to finite lengths, then you
> cannot have an infinite number of them. Finite.
> >
> > 2. Does S have a finite number of elements?
> Same question. Finite.
> >
> > 3. Suppose I define F as the number of elements in S. Is F a
> > finite natural number?
> Yes, it is the largest member of S, which doesn't exist, but which is in the
> set by definition. F does not really exist for the entire set. For any set of
> naturals for which the size exists, the largest member is at least that large.
> So, IF you define F to be the size, THEN it is the largest member of the set,
> BUT there is no largest member of the set, SO F does not exist.

So you have some definition of finite set which includes
sets whose size is an existing number.

I don't suppose you can see why this definition of "finite
set" confuses us poor slobs who think "finite number of
elements" means "number of elements equal to a finite
number".

> > 4. Is F in S?
> Yes, it is the largest member of S. Same as above.

But it doesn't exist. So the size of this finite set is
not finite.

> > 5. Is F equal to 10^1 + 10^2 + 10^3 + ... where I have a term
> > 10^k for every finite natural number k?
> That is the formula, yes.
> >
> > 6. Is there a term 10^F in the expression 10^1 + 10^2 + 10^3 + ...?
> There is always a finite larger than any particular finite you name. You know
> that. Nice try.

What does that answer have to do with the question? I
didn't ask about largest members, I didn't say anything
about the successor to F. I just asked if whether
"F is a member of S" means that a sum of terms like
10^k where k is a member of S includes a term 10^F.
That's a sum over members of S. It has a term for
every member of S. F is a member of S. So it has a
term for F. Yes?

> > Note that I'm not claiming there is a last natural number,
> > or that it is F. All of the above are simple yes or no
> > questions.
> If F is the size of the set, then it is the largest finite, which doesn't work.

Which doesn't matter. I don't care if it's the largest.

I just want to know if "S is a finite set" means that
its size is a finite number, i.e. is in S.

> There is no identifiable size of the set of finite naturals.

So its size is NOT a finite number?

> > I would think that the answers to 1, 2, and 3 are the
> > same, and also that the answers to 4 and 6 are the same.
> > But I have no idea what you think.
>
> 1 and 2 are the same question, as are 3 and 4. 5 is a statement of the formula
> I put forth, and 6 is, whther you admit it or not, the "largest finite"
> argument.

There was no "largest finite" argument. These questions
are only dealing with the question of set membership.
Is the size of a finite set in the set of finite numbers?
That's all I'm asking.

You seem to be saying:

Yes. It's finite, so it's in the set of finite numbers,
but also

No. It doesn't exist, the set of finite numbers doesn't
have a size, so the size is NOT a finite number.

- Randy

.

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