Re: abundance of irrationals!)


aeo6 Tony Orlow wrote:
> > > When you wrote this, I had already posted two proofs.
> >

Your "proofs" (typos corrected):

Proof 1:
As we enumerate the naturals starting at one, at each step in the
enumeration
we are adding the successor to the largest element, thereby
incrementing not
only the set size, but the size of the largest element. If we do this
an
infinite number of times to achieve an infinite set, we have not only
incremented the set size an infinite number of times, but the largest
element as
well. So any infinite set of natural numbers must have infinite
members, since
every time we increment the set size we increment the range of values.
QED

Your "procedure" for incrementing a set is an unending one: there is no
normal meaning to "completing" an unending procedure, and (as many
people have already pointed out) infinity does not get "achieved" in
this way.

Also, spot that your "proof" essentially talks about the infinite
sequence of initial subsets of the natural numbers. Notice how you
mention the largest element of each of these subsets, and recall that
previously you explicitly mentioned the role of this largest element as
the "salient point". Notice that you are claiming that because
something holds true of each of these subsets, it 'must' hold true of
the limit of the sequence, which is the whole set of natural numbers.
But here is a transformation of your "proof":

Proof 1A:
As we enumerate the naturals starting at one, at each step in the
enumeration
we are adding the successor to the largest element, thereby
incrementing the
set size, and creating a new set which definitely has a largest
element. If we
do this an infinite number of times to achieve an infinite set, we
have not only
incremented the set size an infinite number of times, but have created
an
infinite set of natural numbers which has a largest element. So any
infinite set of
natural numbers must have a largest member, since every time we
increment the set
size we create a new largest member. QED

But even you don't actually believe that there is a largest natural
number. So the "proof" is (very obviously) bogus.

Proof 2:
We define the naturals by strings of digits. In any digital system of
number
base n, in a string of d digits we can represent n^d different values.
If we
generate an infinite set, then n^d must be infinite. If n and d are
both
finite, then n^d is finite. We can only have infinite n^d if either n
or d is
infinite. We know that in digital systems n is always finite (base-10,
base-2,
never base-oo), so in order for n^d to be infinite, so that the
digital system
can represent an infinite number of values, d must be infinite, which
means we
have an infinite number of digits required to represent all the
values. Since
we know that in a digital number system, an infinite number of digits
to the
left of the digital point (remember we're talking about natural
numbers)
represents an infinite number. So, for an infinite set of naturals we
require
an infinite number of digits, which means we have infinite values in
the
infinite set. QED

Everywhere you are muddling up the size of individual elements of the
set with the size of the set. Yes, to write down all of the natural
numbers in a list requires an infinite number of digits, because there
are an infinite number of natural numbers. There is not an individual
natural number that has an infinite number of digits, because we DEFINE
natural numbers as being finite natural numbers.

Look, when I say consider the set N of finite natural numbers, being
the set including all finite natural numbers, *each* of which can be
written with a finite number of digits, and absolutely not including
any numbers not satisfying this condition, am I ok? _Can_ I consider
this set - Yes, or No?

If I may consider it, I want you to tell me whether this set, my set,
the one I've called N, has a finite number of elements or not. Remember
that "finite" means you can count them by reciting a ditty, and knowing
that this ditty will eventually stop, and the name you stop on is what
we call the size of the set.

I think you have in fact already agreed that this set, my set N, not
any other set, does not have a finite number of elements. Right: this
is what mathematicians call an "infinite set". If you want to, choose
any other name you like (like "unlimited set", "almost infinite set",
whatever, distinct from 'finite'), tell me what you want to call it,
and I'll use the same name. The only thing you are not allowed to do is
tell me that although I have stated clearly what set I am considering,
you think I'm really considering, or ought to be considering, or
something, a quite different set. OK?


> Try commenting on what I said above, before you wrote all this.
Consider the
> least infinite to be an upper bound on all finite values. If the set
has an
> infinite size and all finite element values, then the size is greater
than any
> element, and there is not one which has a size as large as the set.

This seems to be correct. If you like, oo is an upper bound all of the
natural numbers, the set of natural numbers (my N above) has an
infinite size (cardinality is a better word, but leave that debate
separate), and all elements (natural numbers) are finite, and none of
them is as big as the size of the set. Is this you, Tony? You seem to
have got there?


> > "1000..." isn't even a 10-adic. 10-adics do have reasonable
arithmetic:
> > you can add them, for example. Have you thought how you would add
> > 100... and ...111 ?? Beware the slippery slope here: Defenders of
the
> > Infinite Integers do tend to end up with ever increasingly
ill-defined
> > patterns of digits and dots.
> I would add them to 2111111...... It is possible to add any two
numbers of this
> sort, as long as their endpoints are known, I believe.

I'm afraid your belief is (once again) based on the false intuition
that 'infinity' is the place you get to at the 'end' of the natural
numbers. The 10-adics have an *unending* string of digits to the left.
An *unending* string of digits to the left does not have an _endpoint_
at the left, because *unending* means "not having an end(point)".

The problem comes with
> digits "somewhere" in the middle. For instance
9999.....9956479999...99999 is
> not a distinct number, since that non-9 string in the middle is not
in an
> identifiable location. Are you sure 100000..... isn't 10-adic? I am
no expert...

<snip courtesy the scissors of mercy>

What did I say about the slippery slope and
"Any-old-pattern-of-digits-and-dots"?

Brian Chandler
http://imaginatorium.org

.

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