Re: Dial 999 for the real number line

On Tue, 29 May 2007 10:22:47 +0100, Six Letters wrote:
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.

Functions can be continuous. Sets are not continuous, but they can be
connected, which the reals are. They are also complete, which means they
satisfy the least upper bound property. But what do you care? There are
no real numbers in your world.

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.

No, for the n-th time, you are misusing the word "sequence". A sequence
is a mapping whose domain is N, the set of natural numbers. Since there
is no N in your world, there can't be any sequences.

<snip>

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.

Not at all. You simply have failed to understand the consequences of
your assertion that infinite sets do not exist.

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.

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


--
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/>
.

Relevant Pages

  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... fis the position in the old sequence from which you get the n'th ... For infinite sets of labels, ... > primarily about the cardinality of the continuum and continua, ... > cardinality, and that it could be explained that way. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... Since "the continuum" in mathematics simply means the real line, ... For this is where set theory tries to get to grips with sequence. ... existence of an infinite set. ... Identity Paradox. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... Since "the continuum" in mathematics simply means the real line, ... axioms. ... For this is where set theory tries to get to grips with sequence. ... existence of an infinite set. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... Paradox, which you have so far airily dismissed as nonsense, and simply ... an infinite decimal expansion is ... 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. ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... ....2222, they are actually infinitely distant elements of a sequence, ... smallest positive number, on the infinite scale. ... That's the ordering used by Cantor to prove their countability. ... If a set S is countable, then there is a total order < of S such that ...
    (sci.math)
Quantcast