Re: Dial 999 for the real number line

On Mon, 28 May 2007 23:24:48 +0000 (UTC), Dave Seaman
<dseaman@xxxxxxxxxxxx> wrote:

On Mon, 28 May 2007 15:36:49 +0100, Six Letters wrote:
On Sun, 27 May 2007 00:26:14 +0000 (UTC), Dave Seaman
<dseaman@xxxxxxxxxxxx> wrote:

You see a paradox. I don't.

Fair enough. But it seems to me that the impossibility of
approximation, as I have described it, is at odds with the notion of a
continuum.

Since "the continuum" in mathematics simply means the real line, that
statement of yours is self-contradictory.

Lol. Well, that would cover your arse. But isn't the continuum
supposed to be, er, continuous? Oh, I get it. 'Continuous' means
real-number-line-like.

The axiom of infinity simply says that there is a set that contains 0 and
also constains the successor of each number in the set.

You don't accept that. Fine. There is nothing left to discuss.

Well, there's that, and also the bit about ZF being completely
fucked.

You have not presented any evidence of that.

To the end constraining the argument to the shape of your beloved
axioms. You were right, though, to drive me towards the axiom of infinity.
For this is where set theory tries to get to grips with sequence. To the
extent that it succeeds in doing that, then it fails to assert the
existence of an infinite set.

Are you claiming that the axiom of infinity fails to assert the existence
of an infinite set?

Yes As I understand it, it defines a sequence, which parallels the
concept of a natural number.

Wrong. "Sequence" is a defined concept and appears nowhere in the
axioms.

All I meant is that what it asserts the existence of, is in fact a
sequence. But it dawns on me that all my arguments have in a way been
indirect. I am arguing that a sequence is not infinitely extended, and
therefore N (or its 'axiom of infinity' counterpart) cannot be the infinite
set that set theory requires. You, secure in your knowledge that N can be
reduced to sets, are having none ot it, and cannot take such arguments
seriously. Perhaps this explains why we seem to be arguing at cross
purposes so much. And I accept that my position is grossly incomplete
without backing up this claim that sequences do not reduce to sets, and
that I need to tackle head-on concepts like successorship. If my arguments
about sequences have weight, then there must be something wrong with this
reduction, but I have not demonstrated what it is. And I confess I am not
sure how to go about this. One immediate thought is the following:
If a (finite) ordinal is (roughly) the set of all ordinals less
than itself (Have I got that right?), then how do ALL the ordinals get
defined, if there is no last one? Don't we have to assume already that the
(finite) ordinals are a completed, i.e. infinitely extended set? Which
would very much beg the question, from my point of view. But I need to give
this more thought and some study.

If a set contains 0 and also the successor of each of its members, then
the set cannot be finite. Proof by induction: the set contains more
than 0 elements. And if the set contains more than n elements, then it
also contains more than n+1 elements.

If a set does not have the size of any natural number, then the set is
infinite. That's what the word means.

I do not believe set theory has propietorial rights over the
meaning of 'infinite', but in the light of the above, I will defer taking
up thess points properly.

I don't mind if you tell me that you think the set N does not exist, but
don't try to tell me it is not infinite. Sets have to exist before we
can talk about whether they are finite or infinite. Make up your mind.

But doesn't it contain all the natural numbers?

Are you talking about the set described by the axiom of infinity?

Yes. Or just the natural numbers, considered as a set.

Which in your world does not exist, right? Nevertheless, in any world in
which that set does exist, it is infinite. If the set does not exist,
then we can't even ask the question of whether it is finite or infinite.

<Much repetition snipped.>

If you post again, please avoid mentioning natural numbers, sequences, or
real numbers. None of these exists in your world.

That is a gross distortion of my position.

I have work to do. Thanks for making me realize that. In the
meantime can I offer you another little paradox?: I will call it the
Identity Paradox. Please give a better reason for it not being paradoxical
than the one you gave for the Approximation Paradox.

Let A be any infinite sequence. It scarcely matters what.

B is an infinite sequence.

B is the same as A in the first position of the sequence, but it
(B) is not (the same as) A.

B is also the same as A in the second position, but it is not the
same as A.

B is also the same as A in the third position, but it is not the
same as A.

If this goes on forever, is B the same as A, or is B different from
A?
If B is the same as A in every sequence position, then B must be
the same as A, and the bit about not being the same as A becomes nonsense.
But every sequence position is finite. Therefore it is always the
case that B is the same as A up to and including the nth position, but is
not the same as A.


Regards, Six Letters

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