Re: Dial 999 for the real number line

On Sun, 6 May 2007 19:59:35 +0000 (UTC), Dave Seaman <dseaman@xxxxxxxxxxxx>
wrote:

On Sun, 06 May 2007 16:49:07 +0100, Six Letters wrote:
On Tue, 1 May 2007 13:44:03 +0000 (UTC), Dave Seaman <dseaman@xxxxxxxxxxxx>
wrote:

On Tue, 01 May 2007 13:46:22 +0100, Six Letters wrote:

Given one irrational, say pi, there is obviously a kind of spread
to other irrational. Thus if:
.141592.......... (pi expansion)
is a number, then so is:
.131592......... (continuing as pi expansion),
changing the 4 in the 2nd decimal place to a 3.
And generalising, all finite permutations from the integers 0 to 9
length n as initial strings followed by the expansion of pi after the nth
decimal place, these are all good, paid up members of the real club too. We
can make n as large as we like, so one irrational number generates an
infinite number of others.

But there are only countably many real numbers that can be specified by a
formula or an algorithm, leaving uncountably many that cannot be so
specified.

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

One of those permutations could of course be the initial decimal
expansion of pi itself. More generally and directly for my purpose, we can
say that any irrational number, pi for example, consists of a finitely
long decimal expansion followed by an infinite expansion. Since the finite
part can be as long as we like, we have an infinite sequence of initial
expansions (already a Cauchy sequence) followed by, in each case, an
infinite expansion. Isn't the second infinite part superfluous?

Is 1/3 a real number? It doesn't have a finite decimal representation.

I believe it is superfluous, but it is necessary to understand
infinity correctly. Why can we not just posit some arbitrary irrational,
where an infinite decimal expansion is as it were an infinite collection of
choices for the digits in each place? Because there is not even an actual
infinite expansion for pi; better, what it means for there to be an
infinite expansion here is that there is some method (formula, any device)
for producing successive digits of pi without end.

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.

You are only confusing yourself by dragging in irrelevant considerations.


I can specify the exact value of pi very compactly. Pi is the smallest
positive root of the unique function f satisfying the initial value
problem:

f + f'' = 0
f(0) = 0
f'(0) = 1

It's possible to give precise definitions of some numbers (even
transcendentals) without mentioning decimal digit strings.

The notion of infinity as without bound, limit or end, as radically
sizeless, which is the real lesson of the Galileo paradox, is facile and
cardinal. (Contrast the notion of infinite sets being equinumerous with
proper subsets, which I feel will have future historians of mathematics
chuckling for decades.) That there exists an infinite capacity to extract
the digits of pi, does not mean there literally exists an infinite decimal
expansion of pi. On the contrary, there is literally no end to the
specifying of the digits of pi.

What does it mean for a mathematical object to "literally exist"? Does
the empty set "literally exist"? Where can I find it? Keep in mind that
there can only be one empty set, by the axiom of extensionality. Does it
reside in our galaxy, the Milky Way? Wouldn't that be an amazing
coincidence? Does that mean that civilizations in other galaxies cannot
do mathematics?

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

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? 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).

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).

On which line of your table can I find the number 1/3?

To say that the list only produces rational numbers is like saying
the Leibnitz formula (for example) for pi only produces approximations to
pi.

What you are trying to say here is that the numbers in your table are
dense in the reals. That's true, but it does not change the fact that
the set of numbers in your table is not identical to the set of real
numbers. An accumulation point of a set need not be a member of the set.

Any cut-off produces a rational number in either case, but the Leibnitz
formula is a formala for pi, not for approximations to pi. And the list of
reas is all of them (or if you like, there is no 'all' in such a case).
There are no other reals at the end of or besides this list. Indeed the
decimal system is a tool for constructing rationals. To think it incomplete
is to imagine some reality beyond the model. The model is the reality. So
where are these real ratios or distances, like pi or sq.rt of 2, in this
list? They are there, in the only way they could be, in the endless
successive approximations ocurring there.

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

It doesn't.

Your table contains only binary digit strings, not formulas. On which
line of your table can I find pi? I will accept pi-3, since that is a
number in [0,1].

I should have been more explicit. The proposal is that some
numbers, irrational numbers in particular, do not exist as individual
members in this list, but as embedded sequences.

Yes, you should have said that. It would have been even better if you
had simply said that your table of numbers is dense in [0,1]. That would
be much clearer.

(Incidentally, since what
is at issue is infinite decimal expansions, infinite 0s included, the table
should more accurately have simply omitted the trailing 0s). So, reverting
to decimal (binary was chosen for compression rather than perspicuity), pi
-3 would be:

.1
.14
.141
.1415
etc.

The important thing, I am suggesting, is that there should be
something driving this infinite sequence, some formula or rule or pattern.
Arbitrary infinite decimal expansions are a mirage.

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.

For rational numbers, which in the unit interval will be proper
fractions, we can always choose some number base in which the expansion
terminates -- trivially and at least the base which is the whole number the
denumerator. However we can also regard for example 1/3 in the binary table
as being represented not by a single member, but by an embedded sequence,
namely:

.01
.0101
.010101
etc,

(It is of course .<01> recurring.)

Yawn. Another long-winded way of pointing out that your table of numbers
is dense in [0,1].

To believe in some arbitrary real, an infinite number of choices
for the decimal places, is like thinking one can count to infinity. This is
not a matter of ordinary human limitation. What's laughable about counting
to infinity (1,2,3.........) is that there is no end to it, so beginning is
misconceived. Be immortal, and your unceasing years will simply match your
unending task. Nor is it a matter of importing some notion of process
(first you have to say 1, then 2,..). It is simply that there is no end to
the business of making an infinite number of choices, sequentially or
simultaneously, and therefore, again, no real beginning either.

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.

Is your real objection simply that you don't think infinite sets exist?
In that case, you don't accept the natural numbers, let alone the
rationals or the reals. Why don't you just say what your real objection
is?

Because I didn't start out with any such finitistic assumptions. In
a sense, as a summary of my point of view, yes, I don't believe infinite
decimal expansions exist. But the point is to get at the meaning of such
decimal expansions, rather than to rule them out by fiat.

Then I will ask again: do you accept the axiom of the power set?

Cauchy sequences are of no help. Of course, GIVEN some sequence, we
can prove that if it's Cauchy then it converges (and vica versa). But we
are not GIVEN a sequence by some arbitrary real. The Leibnitz formula gives
us a sequence. Any sort of pattern or rule will give us a sequence. But the
sequence that consists of an infinite number of arbitrary choices is a
mirage. Similar considerations apply to Dedekind cuts.
As far as I can tell, the reals so described would still constitute
technically a complete ordered field.

Wrong. The computable reals are an ordered field, but they are not
complete. That is, there is a set of computable reals that is bounded
above, but for which the least upper bound is not computable.

That would be the conventional point of view. You introduced the
notion of computable reals, though. I'm talking about the reals per se. It
would be a consequence of my point of view, I think, that there are no
unspecifiable reals. I'm not sure whether computablility adds anything new.

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.

I appreciate your replies.

Thanks, Six Letters
.