:: divergent numbers ::
When I visited the Earlham College campus in Indiana some months ago, I was privileged 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 likewise has infinite digits.
What it takes to be "specific" is simply an algorithm. The notion of "convergence" as getting smaller and smaller (closer and closer) 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).
In that case, if each is unique, then may we write: 314159.... (pi's digits) > 161803... (phi's digits)? What about 999123... > 55555... > 1239... ? Some will say this expression is meaningless, because these are meaningless strings masquerading as signifying numbers.
Would making the rule for extending them help at all?
The use of the inequality sign above violates our sense of a "unique infinity" (or do we have that sense?). The idea that 999... (always 9s) has a "permanent head start in its most significant digit" over 333... (always 3s) seems more of an argument that < and > should go away with "numbers" written like this.
Ordering is not defined, merely equality and inequality? If that's accepted, then these are not members of N as N has a well-defined notion of ordering right? There's never doubt about which element is greater, given two elements. Is that so by theorem or by axiom I wonder?
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 likewise has infinite digits.
What it takes to be "specific" is simply an algorithm. The notion of "convergence" as getting smaller and smaller (closer and closer) 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).
In that case, if each is unique, then may we write: 314159.... (pi's digits) > 161803... (phi's digits)? What about 999123... > 55555... > 1239... ? Some will say this expression is meaningless, because these are meaningless strings masquerading as signifying numbers.
Would making the rule for extending them help at all?
The use of the inequality sign above violates our sense of a "unique infinity" (or do we have that sense?). The idea that 999... (always 9s) has a "permanent head start in its most significant digit" over 333... (always 3s) seems more of an argument that < and > should go away with "numbers" written like this.
Ordering is not defined, merely equality and inequality? If that's accepted, then these are not members of N as N has a well-defined notion of ordering right? There's never doubt about which element is greater, given two elements. Is that so by theorem or by axiom I wonder?
I propose we name this "set of infinite serial numbers" (which have many properties in common with members of N) the "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 infinitely digits 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, if these are actually distinct sets. In which set should we put 22222.... (always 2).
Given N is defined to be infinite, it follows that it *must* have room for such numbers with an infinite number of 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).
[first draft posted to mathfuture earlier same day]
For further reading:
Given N is defined to be infinite, it follows that it *must* have room for such numbers with an infinite number of 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).
[first draft posted to mathfuture earlier same day]
For further reading:
