Dial 999 for the real number line


Or 911 for the continuum


Take two reals, expressed as decimals, in the usual unit interval.
We can make them as close together as we like. The difference between them
(subtraction) can be vanishingly small. It will though, presumably, always
be finite. Let the two reals be:

A = SnP
B = SnQ

where Sn is the initial decimal expansion, length n (decimal places), and
P, Q are the expansions from the n + 1 th decimal place.
n +1 is where the numbers A and B first diverge. It doesn't matter what
happens after that. It must be that n is finite, so that however large it
is, it's size pales into insignificance compared to the infinite expansion
after n. In this sense A and B are infinitely different from one another.

It is no help that there is always another number, C, between A and
B. A and B were arbitrarily chosen. A and C or B and C will themselves be
infinitely different from one another, as will D, between A and C, be from
A, and so on. It's spikes all the way down.

Is this the continuum? It looks to me more like an
electrocardiograph.

Once tou have one irrational number, pi for example, then you seem
to have them all, like a lamp lighting up a vast, dusty cavern. If pi can
go:
(3).1415926,,,,,,, for ever,
then surely you can have old decimal expansion going on forever,
representing some other irrational number. Or can you? Maybe the
specifiable irrationals -- ones for which there is some formula, or any old
pattern or rule dictating the decimal expansion (or continued fraction
expression, which can then in any case be fed back to the decimal
expansion) -- are more like a pencil-thin beam, visible only by the
surrounding darkness.

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.
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?

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.

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.

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).

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

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.

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.

I think this understanding of infinity also has a bearing on Zeno,
probably the most fundamental of the paradoxes about infinity, but that
takes us rather far away from the nominated subject.

Forgive any polemicism or didacticism in tone. It's just easier to
express one's point of view without equivocation. I'm happy to see this
ripped to shreds, if done righty.

Six Letters





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Relevant Pages

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  • Re: Dial 999 for the real number line
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  • Re: Dial 999 for the real number line
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  • Re: Dial 999 for the real number line
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