Nonsense Numbers?

[kirby urner]

When I visited the Earlham College campus in Indiana some months ago,

I was privilege to address the Philosophy Club, somewhat extra-curricular

yet affiliated with the philosophy department.  I briefly went over a discussion

we'd been having on math-teach (The Math Forum / Drexel) regarding

the following:

We all know about Cantor's work showing N and R belong to different

orders of infinity.  We can make elements of N (1,2,3...) pair with all the

rationals Q by a well defined process, but we can show that no plodding

method forward will map all of R with members of N.

But is N itself "numerable" in the Cantorian sense?  Consider what I call

"mirror pi" which is just the digits of pi-to-the right "reflected in the mirror"

3.14159... -> ...951413

The 3-dots (...) signify that the digits go on forever per known algorithms,

in both cases.  On the left, we say we're converging to some R.  On the

right, I'd say we're "diverging" to a specific element in N, which is likewise

infinite (enough room for mirror-pi?). 

What it takes to be "specific" is simply an algorithm.  The notion of
"convergence" as getting smaller and smaller is distinct from the concept
of "specificity".  Think of serial numbers without ordering (no > or <,

only == and !=).

Or we could simply write pi like this:  314159... with the understanding

that we'll never make use of a decimal point.  We'll just keep writing a

longer and longer string of numbers.  One may imagine some quantity

getting bigger and bigger, but think of it instead as writing the unique

(but infinite) serial number for some grain of sand on the beach, a
member of R (right?), and in this case also a member of N (a positive

integer).

I think you'll find a lot of die-hards wanna keep numbers like 314159...

out of N, because leaving them in messes with N and R having different

aleph numbers.  We also want to keep the idea of "unique infinity" in some

way i.e. if all these "infinite serial numbers" are both truly infinitely big
(the reciprocal of infinitely small) and yet are each "specific to one
element in the set R" (true in case of pi in R, but argued about. with
respect to mirror-pi), then we're left wondering if we can write:

999123... > 55555... > 1239... (all masquerading as real numbers,
but rules for extending them unclear).  That violates our sense of

"unique infinity".  The idea that 999123... has a "permanent head

start in its most significant digit" seems more an argument that

< and > should go away with numbers written like this.  Ordering

is not defined, merely equality and inequality.  If that's true, then

these are not members of N as N has a well-defined notion of
ordering.  So is that so by theorem or by axiom I wonder?

I propose we name "the set of infinite serial numbers" (which have

many properties in common with members of N) as "nonsense
numbers" which "diverge to a unique significand" (that object

which the number symbolizes -- we're taught to think some

name->object model applies when it comes to ordinary pi in
any case, a standard part of the mental baggage).

It will be difficult to distinguish Nonsense Numbers from actual
members of N however i.e. how would we know for sure? 

Given N is defined to be infinite, it follows that it *must* have room
for such numbers with infinity digits (if not, N is finite, a contradiction),
so that really argues for the Nonsense Numbers being a subset of N. 

Perhaps that's how we should teach them then? 

"N is for Nonsense Numbers" (otherwise known as Natural Numbers,

only some of which have finite digits).

Kirby