Re: Dial 999 for the real number line

On Tue, 08 May 2007 14:47:01 +0100, Six Letters wrote:
On Sun, 6 May 2007 19:59:35 +0000 (UTC), Dave Seaman <dseaman@xxxxxxxxxxxx>
wrote:

There aren't uncountably many anything. One of the important
consequences of this way of understanding infinity is that the distinction
between countable and uncountable is undermined. There is only one
infinity.

Do you claim that there is a bijection between the naturals and the
reals?

It's a good question. Firstly, on the subject of set theory. Some
rather insulting person on a thread about Galileo's paradox some months ago
(a thread from which I had to break off -- real life, and all that) kept
harping on about Peano. I've no doubt that Peano is very interesting, even
important, but the idea that it has any reductive thrust I find, nowadays,
rather quaint. It does not begin to address the Galileo paradox, but loses
it in a mass of technical detail.

The Galileo paradox simply points out that the natural numbers are
equinumerous with a proper subset, namely the set of squares. It isn't
really much of a paradox. Exactly what is there about this observation
that you think has not been addressed?

I do not accept that set theory has some
hegemonical provenance on the subject of number. If it could be proved that
there is no bijection between real and rational (yes I know that you
believe it has been proved: it's not that I think Cantor's proofs are
simple-mindedly wrong, but that I believe there is an alternative
conception of infinity, still a work in progress, according to which they
are invalid) it would not have the consequences you think.

You can't get rid of inconvenient facts by changing definitions, but only
by changing axioms. Which axiom(s) of ZF do you propose to change, and
why? Keep in mind that whatever system of axioms you come up with will
not in any way invalidate the theorems of ZF, including the
uncountability of P(N).

There would not
be the humungnous quantity of transcental numbers you thing there are.

You can't make uncountable sets countable by changing definitions. It
would be different if you could find a logical error in each of the many
proofs of uncountablility of the reals, but that is not going to happen.
The proofs are entirely too simple and well known to contain any flaws.

Ontologically, if you like (well I know you don't like, but I will come
back to the question of mathematical existence in a minute) there are only
rational numbers. One can construct irrational numbers from them. You can
put them back on the real number line, if you wish, but this is to obscure
the difference between them. The so-called real number line is something
confused between it's arithmetial and geometic identity.

Are you claiming that Dedekind cuts do not exist?

You will never understand real numbers until you get over your hangup on
decimal representations. A real number is an equivalence class of Cauchy
sequences of rationals. Or, a real number is Dedekind cut. Either
definition will do, and neither makes any reference to decimal digits.

I don't believe I have a hang-up about decimal representations. If
I have a hang-up, it's about infinite sequences. My idea of real nuimbers
sits quite well with the Cauchy definition, except that I don't believe
arbitrary or unspecified sequences are well defined.

Your real problem here is with the axiom of the power set, as I have been
hinting. We don't have to "define" each member of a set individually in
order to show that the set is uncountable. The set of Dedekind cuts can
be viewed as a certain subset of P(P(Q)), the power set of the power set
of the rationals. In fact, using a slightly modified definition of a
cut, we can construct the cuts as a subset of P(Q).

I quite agree we should not take mathematical existence at face
value, which is precisely what I am trying to avoid.

My point is that mathematical existence has nothing to do with physical
existence. Claiming that something has no physical existence is
inconsequential in mathematical arguments. All that matters is what
follows from the axioms.

Pi exists, whether we can write down all its digits or not. See the
definition above. A decimal representation is just that, a
representation. It is not the same thing as a number.

Your criticism here is curiously inept. Of course pi exists, is a
real number. What I am asking is whether the infinte decimal expansion
(.14159.............) exists; or rather what is the meaning or status of
such an infinite expansion.

Every real number has an infinite decimal expansion. It exists, whether
we can write down all the digits or not.

There's the rub. Who now is taking mathematical existence at face
value?

You are claiming that the decimal representation of pi has no physical
existence. I agree. My point (see above) is that this is irrelevant to
the question of whether pi has mathematical existence. It follows from
the axioms. Perhaps that is what you mean by taking mathematical
existence "at face value". If existence is not determined by the axioms,
then what we are doing is not mathematics.

Actually I would agree that a real number has an infinite decimal
expansion. But I would give a gloss on what that latter phrase means,
according to which it does not follow that any infinite decimal expansion
is a real number (because an arbitrary infinite decimal expansion is
ill-defined).

Which digit position in the representation of pi do you consider to be
ill-defined? In principal, any specific digit can be computed, given
enough time and space.

You are committing two basic errors here. First, you are claiming that
the decimal representation of pi does not exist, simply because no one
has written it down. Non sequitur. Second, you are claiming (I think)
that pi is not well defined unless its decimal representation can be
written down. Again, non sequitur. Real numbers have other
representations besides decimal digit strings, and I gave you one that is
perfectly well defined.

By the same token, the systematic listing of the reals in the
following table (in binary for convenience; -> means trailing 0s):

.0 ->
.1 ->
.0 1 ->
.1 1 ->
.0 0 1 ->
.0 1 1 ->
.1 0 1 ->
.1 1 1 ->
etc.

such a table represents all the reals (in the unit interval).

I repeat my question. On which line of your table does the number 1/3
appear?

It doesn't.

Do you accept the axiom of the power set?

See above. The power set axiom seems fine for finite sets. Since
the set theory understanding of infinity is fundamentally flawed, and I am
not expert in all the interdependencies of axioms and theorems of set
theory, I could not venture more.

The power set axiom is not needed for finite sets. The axiom has content
only when infinite sets are considered.

A set is called "finite" if it is equinumerous with some natural number.
A set that is not finite is called "infinite". Please explain where you
think you see a flaw in this definition.

You seem to think that the axiom of infinity is the one that is upsetting
your intuition, but you are mistaken. It's the power set axiom, not the
axiom of infinity, that you are having trouble with. If we accept all
the axioms of ZF except the power set axiom, then we can prove that
infinite sets exist, but we can't demonstrate any uncountable sets.

There is no axiom that says we can count to infinity. There is, however,
an axiom that implies the existence of an infinite set. There is a
difference.

Again, it's a good point, but again I think you are taking
'existence' at face value. I agree that the set of natual numbers 'exists',
but this is because there is a rule that generates it (essentially, add 1),
but that does not mean that any old arbitrary infinite set you like exists.

I take sets to exist when it is implied by the axioms. If that is what
you mean by taking existence "at face value", then I plead guilty as
charged. You are free to invent your own axioms, but you are not free to
dictate to other people what can be concluded from the axioms of ZF.

What do you mean by a "specification"? If a number can be specified by a
formula or an algorithm, then it is a computable real. Is it possible to
"specify" a real without providing a formula or an algorithm?

Probably not, but I am not sure of the relevance.

Actually, I can think of ways to do so, but I was trying to find out what
your point was. Both the computable numbers and the "specifiable" numbers
(by any reasonable definition) are countable sets, and therefore neither
can be a complete ordered field.


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

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