Re: infinity

Tony Orlow says...

>Is aleph_0 the number of finite natural numbers?

Yes.

>Is there a sequence of them that is aleph_0 long?

Yes.

>Does each one have an order in the sequence?

Yes. aleph_0 does *not* appear in the sequence.
The only ordinals that appear in the sequence
are ordinals that are *less* than aleph_0.

>Are there elements which have no natural number index?

No. Every element has an index that is less than aleph_0.

>Is it true and provable inductively that each successive
>natural starting from 1 has, as its value, its order in
>the set, such that 1 is 1st, 2 is 2nd, etc?

No. 0 is the first ordinal. 1 is the second ordinal, etc.

>At what point in the succession of ordinals do we switch
>from using successive ordinals to limit ordinals?

aleph_0 is the first limit ordinal. If an ordinal is smaller
than aleph_0, then it is not a limit ordinal. aleph_0 has
the property that

x < aleph_0
->
x+1 < aleph_0

>Does it require a finite or infinite number of steps before we switch?

Yes, aleph_0 is an infinite ordinal, it cannot be reached in a
finite number of steps.

>What kind of an ordinal does not correspond to an
>ordered place in the set?

That question doesn't make sense: every ordinal has a place;
the place for ordinal x is after all the ordinals that are
smaller than x.

>> Subtraction is not well-defined for limit ordinals.
>
>At what point does subtraction break down.

If an ordinal is a successor ordinal, then it has a predecessor.
If it is a limit ordinal, then it doesn't have a predecessor.

>But you generate, supposedly, an infinite number overall,
>one increment and one element at a time. You repeatedly add
>1 to the value you just got by adding 1. This ammounts to
>adding an infinite number of 1's.

>> I am saying nothing of the kind. I am saying there are an infinite
>> number of elements in the sequence, each of which being only finitely
>> far away from any other.
>
>But, the elements are all in a sequence, are they not, so that
>every element is either before or after any other element? If
>you have such a set with n elements, doesn't that mean that
>some element is n-1 elements after some other?

If n-1 is defined, yes. If n is a successor ordinal, then no.
In that case, there is no n-1.

>Certainly this is true in the finite case. I see no reason
>to abandon this notion of order in the infinite case.

Well, that shows the difference between what you are doing
and actual mathematics. In real mathematics, we define our
terms and see what the consequences of those definitions are.
In contrast, you make up your consequences as you go along.
It's not mathematics if you are just giving your opinion.
It becomes mathematics when you can *prove* your conclusion
from your definitions.

>> Not at all, that is a correct statement as well. There are an
>> infinite number of successors for any given number, each of which
>> is only finitely far away.
>
>When you say "finitely far away", what in the world do you mean by
>that, if not number of elements in sequence?

It means that for any two natural numbers x and y, |x-y| is a finite
number. What does that mean? It means that the set
{ 0, 1, 2, ... |x-y| } is a finite set. What does *that* mean?
It means that there is no way to order the elements of the set
so that there is no largest element.

Infinite set : there is an ordering on the set with no largest element.
Finite set : every ordering of the set has a largest (and smallest)
element.
Finite ordinal : x is a finite ordinal if the collection of all ordinals
less than x is a finite set.

With these definitions, the set of all finite ordinals is an infinite
set, but for any two finite ordinals x and y, |x-y| is a finite ordinal.

>> Incorrect. The set is of infinite size.
>
>Hand waving proves nothing.

It's not handwaving. It's carrying out the consequences
of our definitions. If you define a set to be infinite
if it can be ordered in a way that has no largest element,
then the set of finite ordinals is infinite.

>How can every value remain finite, when there are an infinite
>number of them in sequence, each equal to their position in
>this infinite sequence, with supposedly infinite indexes?

There are no finite ordinals with infinite indexes.

>Are there infinite indexes?

No.

>If not, what does it mean to say the set has an infinite size?

It means that there is no largest element.

>> It simply comes down to the fact that there are an infinite
>> number of finite numbers. Once you accept that, all of your
>> invalid arguments magically go away.
>Haha. Good one. Resistance if futile. Join us, and all of your
>problems will magically go away. Drink the water. C'mon.
>It won't hurt you.

Well, learning to actually do mathematics certainly won't
hurt you. Rigorous mathematics doesn't force you to accept
*anything*. It only allows you to work out the *consequences*
of a set of definitions.

In contrast, what you are doing isn't mathematics. You
aren't working out the consequences of assumptions, you
are picking consequences you like, and then changing
assumptions and definitions as you go.

--
Daryl McCullough
Ithaca, NY

.

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