Re: Dial 999 for the real number line

On Fri, 11 May 2007 12:57:09 +0100, Six Letters wrote:
On Wed, 9 May 2007 13:12:01 +0000 (UTC), Dave Seaman <dseaman@xxxxxxxxxxxx>
wrote:

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?

Not much of a paradox on that one-sided way of interpreting it. I
would draw a different conclusion: that infinite sets are not numerous,
equi or otherwise.

Is there a bijection between the naturals and the squares?

That is only one half of the paradox. I don't know if you
contributed to the thread about Galileo's paradox I started months ago, but
I'm loathe to go over old ground.

My question directly addresses the issue of whether infinite sets have a
cardinality. But suit yourself. I just looked up that thread and I
don't see anything there that is worth discussing.

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).

What makes you think I want to work in ZF, or any other set theory?
Does set theory hold the keys to the kingdom of heaven?

Pretty much all of mathematics is grounded in set theory.

To your satisfaction, maybe.

It's not just my opinion. Aside from category theory, there just isn't
much in mathematics that isn't defined in terms of sets. If you open a
book on differential geometry or algebraic topology, you may not find any
mention of ZFC, but it's implicitly behind the definitions that you will
find there.

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.

On their own terms, they are. But what set theorists do not like to
admit is that they take a position about infinity, that there might be
alternative ways of understanding infinity, according to which, indeed the
usual proofs do not work.

Nonsense. Set theorists frequently investigate alternative axioms, and
many definitions of infinity have been proposed.

Is that the extent of metamaths, by your understanding? Is it not
possible that all those alternative versions of set theory make the same
mistaken assumptions about infinity?

Which assumptions are those? The assumption that an inductive set
exists? The assumption that unions exist? The assumption that power
sets exist?

I do not think it unreasonable of you to ask
where in its axiom system do I understand ZF to most clearly exhibit these
assumptions about infinity, where according to me it most clearly departs
from reality. But for the moment I would rather try and clarify, and
admittedly get clear in my own mind, my alternative understanding of
infinity. Well or ill, I am doing metamaths, foundations, here, not set
theory.

Set theory *is* the foundation of mathematics.

That's practically a conversation stopper.

See above. Have you taken any courses on the foundations of mathematics?

Let D be the set of decimal digits. There is a mapping f: N -> D such
that f(n) is the n-th digit in the decimal representation of pi. That is
what I mean when I say the decimal representation of pi exists. It's
even a computable function, though computability is not necessary for
existence.

That is a purely formal existence of no interest.

It's of interest to mathematicians. This is sci.math.

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).

Each decimal digit string is a mapping f: N -> D. Given such a mapping,
there is a Cauchy sequence of rationals such that the n-th term of the
sequence is the truncated decimal with n digits.

Fine, that much I have already taken for granted.

Viewing the real
numbers as equivalence classes of Cauchy sequences of rationals, we can
conclude that there is exactly one real number corresponding to the given
digit string.

The "given digit string"? Have we been given a unique digit string?
Which one is it?

Any one you like. I'll call it S.

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.

It's not pi that has ill-defined digit positions, but arbitrary
infinite decimal expansions.

Incorrect. For every digit string S and for every natural number n,
there is a unique digit S_n. That's what it means for S to be a digit
string.

See above. Each infinite decimal determines a unique real number.
Conversely, each real number determines an infinite decimal, which we can
make unique by choosing not to have it end in all 0's.

That's precisely what I disagree with. The traffic is only one-way.
Each real determines an infinite decimal (did you mean it is made unique by
choosing to have it end in all 0s?)

No, I meant what I said. What you suggest is nonsense. Not every real
number has a decimal representation that ends in all 0s, but every real
number does have a decimal representation that does not end in all 0s,
and this representation is unique.

For example, the number 1/2 has two decimal representations, 0.50000...
and 0.49999..., but the rule selects the latter, since it is the only one
that does not end in all 0s. Pi, on the other hand, has only one decimal
representation, and sure enough, it does not end in all 0s.

The converse is not true. Why? Because
on a correct understanding of infinity some concrete meaning has to given
to what it is for an infinite decimal to exist -- they are not just out
there in some Platonic realm.

Can you give an example of an infinite decimal that does not determine a
real number? That's what it means for the converse not to be true.

I have no general objections to infinities in mathematics. One can
contemplate an infinite collection of distinct objects, or in set theory,
an infinite set whose elements have only identity and membership. There
might have been, perhaps, an infinite number of stars in the universe
(might even BE an infinite number of stars if some reflections about scales
of clustering turn out to be confirmed, apparently). Such a set would, as
it were, have infinity supplied, without any 'construction' required (at
least on some facile but quite respectable level).

The axioms of set theory tell us that certain sets exist. It does not
say we can actually construct them.

I understand what you mean, but I don't think you are trying to
understand what I mean by construction.

I mentioned constructible reals earlier in this thread, but you rejected
my attempt. So now I am only discussing the real numbers, not the
constructible reals.

At a stretch, one could
even imagine the Naturals to be like this, though it seems less natural to
do so. (My grasp of set theory may be inadequate here, but it seems that
all infinities in set theory rely on recursive enumeration, but only as a
disposable ladder, so that infinite sets of entirely feature-less elements
may be supposed to exist. One might question that, but I'm inclined to let
it pass for now.)

The elements of a set can't be entirely featureless; they have to be
distinguishable in some sense.

Fine.

With the infinite sequences in decimal expansions, and those
involved in Dedekind cuts and the Cauchy definition of the reals, it is
different. Construction is built in.

Wrong. If construction were needed, we would still be waiting for the
very first infinite set.

A sequence is not IDENTIFIED until, per impossible, it is completed
or finished, OR there is some algorithmic generation of the sequence.

Let me get this straight. Are you claiming that every Dedekind cut can
be identified, but some real numbers cannot be identified? Are you aware
that the Dedekind cuts *are* the real numbers, at least by Dedekind's
definition, which was the first rigorous treatment of the real numbers?

Sets are not even started, let alone finished. Sets simply exist.

But sequences have a beginning, an order, and if they are finite,
an end. So what is an infinite sequence, apart from some generating rule?
Or rather which is an infinte sequence, so to speak? Angels exist on more
evidence. (Did you know you can get uncountably many angels on a pin head?)

A rule is not needed. A sequence is a function whose domain is the
natural numbers. That means it's a set of ordered pairs from NxD (where
D is the codomain) satisfying certain properties. Nothing is said about
a rule.

You can comtemplate all you like, but there is no infinite
precision decimal expansion of pi, because infinities don't end.

Non sequitur.

According to your primitive conception of infinity, no doubt.

When someone accuses you of a non sequitur, the correct response is to
provide a justification for your statement that was challenged, or to
withdraw it.

Still not sure I've put my finger on it, but thanks for persisting.
I should ask you, perhaps, to address the original argument. Are you happy
with an electrocardiographic continuum?

I wouldn't know.

Perhaps you ought to think about it, since it is a consequence of
your understanding of the reals. But hey, who cares about paradoxes when
you've got a set theory you're happy with?

My understanding of the reals has nothing to do with electrocardiograms.



--
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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