Re: Dial 999 for the real number line

On Mon, 14 May 2007 11:04:28 +0100, Six Letters wrote:
On Fri, 11 May 2007 13:55:43 +0000 (UTC), Dave Seaman
<dseaman@xxxxxxxxxxxx> wrote:

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.

My belief is that a set-theoretic treatment of differential
geometry (or whatever) is a set-theoretic treatment of differential
geometry (or whatever).

There aren't many branches of mathematics that don't use the concept of a
"function".

Definition. A function f is a triple (A,B,G), where:

A is a set, called the domain of f,

B is a set, called the codomain of f, and

G is a subset of AxB, called the graph of f, having the
property that for each x in A there is a unique y in B such
that the pair (x,y) is in G.

As I said, pretty much every mathematical object is defined to be a set.
You won't get far in differential geometry without knowing what a
manifold is, or in algebraic topology without understanding homology
groups.

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?

Basic assumptions about the real number system which no tinkering
with the axioms of set theory will save.

Nonsense. No inconsistency has been found in ZFC, and even if one were
found, there is no reason to think it couldn't be fixed. Set theory
survived the crisis of Russell's paradox, after all.

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.

I should have called you on this point before. Mathematics has nothing
to do with reality. Perhaps that is the source of your confusion.

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.

On reflection, I would withdraw that assertion, and say instead
that the mapping does not exist. But to introduce funtions only obscures
the issue, More below.

To introduce what obscures the issue? You mean functions? Oh, so you admit
that functions exist after all?

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.

Have you changed your position here?

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.

No, you give me one. I don't believe in them. We would like to say,
for example: the one beginning .83047753........ But that is like someone
trying to select a cow from a field of black and white cows by saying: I
mean the black and white one. So is it that infinite sequences exist
because I can pick one out, or can I pick one out because infinite
sequences exist? I have a vision of a countless host of angels hovering and
floating around Him who sits on the throne. Theres' one, I'll call him S.

In the first place, it's easy to give a specific digit string. For example, I
choose the one given by f(n) = 1 for each n. This is the string .111111....

It's true that some digit strings (most of them) are not computable and
therefore cannot be "specified", but that doesn't mean they don't exist.

The proof of the proposition "every decimal digit string determines a
unique real number" does not depend on knowing which particular digit
string one has in mind. Quite the contrary, if the proof is specialized
to work for only one specific digit string, then it doesn't apply in the
general case.

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.

If a number can be described (such as pi or e or sqrt(2)), then its
decimal expansion can be computed. It's true that most real numbers
cannot be described, but that doesn't mean they don't exist.

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?)

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.

There are no infinite decimals, so how can I give an
example of one that doesn't determine a real number? What is meant by an
infinite decimal is one that can be generated, one that has its expansion
dictated for it. An arbitrary infinite decimal, which cannot be identified
by enumeration, obviously, and cannot per hypothesis be identified by some
algorithm or rule dictating the expansion, is a fiction.

Part of the study of mathematical foundations involves elementary logic.

The converse of the statement

Every real number determines a decimal digit string. (1)

is

Every decimal digit string determines a real number. (2)

You claim that (2) is false. The logical negation of (2) is

There exists a decimal digit string that does not determine a real
number. (3)

If your position is that decimal digit strings do not exist at all, then
you are admitting that (2) is correct (it is vacuously satisfied) and
therefore you are contradicting your claim that "the converse is not
true."


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.

Fine. I am discussing the real numbers too.

Do your real numbers satisfy the least upper bound axiom? If so, they
are uncountable. If not, they are useless for most of analysis.

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.)

Recursive enumeration is a concept from computability theory. It is not
mentioned in set theory.

One of the axioms of set theory is the power set axiom. The axioms tell
us that P(N), the power set of the naturals, exists. But no method is
provided for enumerating this set.

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.

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?

That is so mangled I wouldn't know whether to say yes or no.

Let's back up and try to establish a few basic facts. Which of these
statements do you disagree with?

(1) The real numbers are uncountable.
(2) The set of Dedekind cuts is uncountable.
(3) The set of equivalence classes of Cauchy sequences is uncountable.
(4) The set of nonterminating decimal digit strings is uncountable.
(5) The sets described in (1), (2), (3), and (4) are all isomorphic.
(6) Any uncountable set contains noncomputable (hence, nonidentifiable)
elements.

According to (5), there is no fundamental difference between the sets
(1)-(4). In particular, if you have identified a member of (2) or (3),
then there is a corresponding element of (1) and of (4), and those are
likewise identified. It's nonsense to claim that one of these has
construction "built in", and another does not. According to (6), each of
(1)-(4) has nonidentifiable elements.

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?

You must be aware that since Dedekind cuts rely on infinite
sequences, we are in no better a position than we are with infinite decimal
expansions.

Not every infinite set is a sequence. A Dedekind cut is a certain pair
of infinite subsets of the rationals, but there are no sequences anywhere
in sight. A sequence is a mapping whose domain is N. Since you don't
believe in mappings, you evidently think sequences don't exist.

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.

I find it difficult to believe you really think this helps. OK,
call a sequence a function from N to S (set), or whatever, then the
question "Which sequence?" simply becomes "Which function?" But since there
is no need to complicate things by introducing functions or mappings, I
will continue to talk about infinite sequences. What does it mean to refer
to a unique, arbitray infinite sequence when there is no way of identifying
it?

Some sequences are identifiable. The identity map is certainly an
identifiable sequence. It's true that there are uncountably many
sequences, and only countably many of them are identifiable, but that
doesn't mean the others don't exist. The point is that we can prove
statements of the form "every S is a T" without ever mentioning any
specific S. We only need to know the definition of what an S is.

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.

Oh come on. The two propositions weren't in simple deductive
relationship in the first place.

Yes, they most certainly were. When you say "A, because B", it means
precisely that A is deduced from B.

I didn't mean to be insulting when I said
your understanding of infinity is primitive. 'Primitive' in some sense is
accurate. But it's not your understanding of infinity, it's a common,
deeply embedded understanding I'm talking about.

But you never explain what you are talking about. Arguing with you is
like trying to nail jelly to a tree. You keep suggesting something is
wrong with the notion of infinity, but you never say what it is. You
claim functions are irrelevant or that they don't exist, but you never
suggest anything to put in their place.


--
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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