Re: infinity


aeo6 Tony Orlow wrote:
> Randy Poe said:
> All elements in the set are finite, so if the size is the value of the largest
> element,

You realize that when you begin a statement "if..." that
what follows might either be true, or not true.

> which doesn't exist but is finite,

If it doesn't exist, then it has no properties.

> then the size doesn't exist but is
> finite.

A sane person would conclude that assuming "the size is the
value of the largest element" leads to contradictions, then
the size is NOT the value of the largest element.

> non-existence doesn't make it infinite.

Non-existence means it has no properties.

> > > > 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?
> Sure, you can have 10^F,

Good.

> but now you are going to say that that is larger than
> the set of all strings,

Don't put words in my mouth. [Remaining words snipped]

Here's all I want to say about that. From the "fact" that F
is finite, even though it is "unidentifiable" and we don't know
which finite number it is, then we have this:

F = 10^F + other stuff.

Any problem with this? You agreed that the expression
10^1 + 10^2 + ... includes a term 10^F, so I can write it
as "10^F + other stuff", right?

And you agreed that that sum equals F?

Do you think there's a finite number F with the property
that it is larger than 10^F?

Note that I *still* haven't said that either F or 10^F
is the "largest finite", nor have I assumed there's a largest
finite, nor have I assumed any property about F except
that:
(a) it is finite and
(b) it is equal to 10^1 + 10^2 + ...

> > I just want to know if "S is a finite set" means that
> > its size is a finite number, i.e. is in S.
> Yes it means the size is finite, obviously. Does it mean the size is a
> particular finite number? No, no more than the largest finite is a particular
> number.

If the largest finite existed, it would be a particular number.

If the size is a finite number, it is a particular number even
if we don't know what that number is. It has a unique successor,
even if we don't know what that successor is. It has a particular
square root, even if we don't know what that is. It is either
even or odd, not both, even if we don't know which. All of
those follow merely from the property "finite number".

If you give me a finite number F, then I can tell you that
there are more than F distinct finite natural numbers. That
is true no matter what F you give me. So the number of
elements in N is more than F, if F is finite.

Even if F is not "identifiable".

- Randy

.

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