Re: Dial 999 for the real number line

On Wed, 30 May 2007 19:42:36 +0100, Six Letters wrote:
On Tue, 29 May 2007 12:05:04 +0000 (UTC), Dave Seaman
<dseaman@xxxxxxxxxxxx> wrote:

You might want to read up on the concept of "omega consistency" to learn
more about this.

Thanks for the pointer. It looks very interesting.
But your statements above and in the previous post, that the
natural numbers and sequences do not exist for me, are simply preposterous.

I meant to say that *the set of* natural numbers does not exist in your
world. Any individual natural number, being finite, presumably exists
for you.

I have explained that the mathematical concept of a *sequence*
presupposes that there is a set of natural numbers. That's why sequences
do not exist in your world. A sequence is a mapping whose domain is N.

The natural numbers exist first as phenomenon, as working
instruments, in advance of any theory that we may have about them. Even the
term 'ordered sequence' has an on ordinary descriptive applicability,
before we set about refininig out some precise technical meaning. It is
clear too that the naturals are infinitely extendable, meaning simply that
we can go on forever (1,2,3,4....).If you are right, then N must be some
kind of set, and a finite set would obviously be useless, so it must be an
infinite (-ly extended) set. Your problem is that you cannot even imagine
that set theory might be wrong.

Yes, I can imagine that. It is theoretically possible that ZF might be
inconsistent. For all we know, there may be a proof in ZF of some
contradiction, say x != x for some x.

You have not produced any such contradiction. In fact, you would be
famous if you did. But the possibility cannot be ruled out.

The arguments I have given for believing
that there are not an infinite number of natural numbers (that infinite
extendability is enough) are independent of set theory. You cannot argue
against my position by just insisting on the truth of your own definitions:
that is not argument, it is simply shouting. It is not true that I am
misusing the word sequence, just because I do accept the set-theoretical
rendition of it, and nor is it the case, as you say, that I simply do not
understand the consequences of my assertion that infinite sets (of the sort
N is supposed to be) do not exist. I understand the consequences very well
thank you very much. And the consequences are that set theory is screwed as
a description or account of the natural numbers, and therefore for every
other kind of number too.

Fair enough, but you still need to produce some evidence of your claims.

The arguments I have used may be poor, but they
are not poor because they contradict set theory, and actually you have not
addressed a single aspect of them. All you do is keep poking a stick out at
me from the behind the protective formalsim of set theory.

You have not presented any arguments. An unsupported assertion is not an
argument.

You find fault because I base my arguments on set theory, but you have
not presented any alternative to set theory as a foundation of
mathematics.

In fact I think I conceded too much when I said my arguments were
indirect. It's just that they are independent. For a while it seemed to me
that there might be some specific place, in the construction of ordinals or
in the axioms employed, where infinite extension is asserted or assumed or
just somehow referred to. It seems to me now that it is just assumed from
the beginning. There is no distinction, in set theory, between infinite
extendablity (such as 1,2,3.. has) and infinite extension. There is
nowhere, as far as I can see, where it says: infinite extendability implies
infinite extension. It doesn't know the difference between them. I
speculated about recursive definition, but I don't have all the necesary
knowledge to pursue it. It was obviously too stupid for you to bother
commenting on. It wouldn't surprise me if there was some deep fault in set
theory which reflects the fact that it has failed to capture the essential
nature of natural numbers. But the best thing I can do is press the case
that assumption of an infinite number of naturals is fundamentally
misguided.

It's easy to model infinitely extendible sets in set theory, but you are
not going to like the way it's done. All that's needed is a sequence of
nested sets, A0 \subset A1 \subset A2 \subset ....

The problem, of course, is that a sequence is a mapping whose domain is
N, and we are back to requiring infinite sets again.

You are free to provide your own definition, of course.

For now, another little sketch.
For you (for set theory) the infinity of the natural numbers is
like an extendable ladder. How could all those lengths of ladder be rolled
out indefinitely, if they were not all ready there, folded up? This, or
something like it, is the guiding picture, the unspoken assumption. But
what if the infinity of the natural numbers was a piece of elastic? There
is no length which is already there. It just expands to any size we need.

Yes, I understand what you are trying to do. But you haven't done it.

Both the ladder and the elastic are equally functional With the former we
can clean the windows of a building of any height, and with the elastic we
can hold up the knickers of a woman of any girth. But what if there were a
building of infinite height (perhaps the Hilbert Hotel), or a woman of
infinite girth? We don't allow that the elastic is infinitely extended. We
can keep stretching and stretching forever, but the infinite woman is going
to have to find some other way of covering her modesty. Are we any better
off with the ladder? The problem is that the ladder starts off in a compact
position, and we using it at the bottom. There is no way of getting to the
'top' floor without going through all the intermediate floors. If there's
anybody up there on the infiniteth floor, they may have to wait a while for
their windows to be cleaned. We're not actually in any better position with
a really infinite extendable ladder than we are with a merely infinitely
extendable type ladder, and in fact, however far we go up, in the former
case, there will always be an infinite number of sections of ladder or
rungs left folded up. Actually the merely infinitely extendable ladders
are much cheaper.

What is your definition of an extendible set that doesn't presume
infinite sets?

It seems to me that your method, whatever it may turn out to be, is
unnecessarily complicated.


--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
.

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