Transfinite theory holds that the set of rational numbers is "denumerable." From my understanding of the theory, I would assume that the set of all namable numbers is also denumerable. I define a namable number as a number that can be named with a finite text string.
I contend that this set is as "denumerable" as the rational numbers for two reasons: The first reason is that finite text strings are finite. For that matter computer scientists have shown that finite character strings can represented with numbers (for example ASCII or unicode). If I converted all of my strings to ASCII, I would have a subset of the natural numbers. Secondly, I contend that the process of naming a number is itself a denumerable process.
Imagine that I was extremely anal retentive. Each time I named a number, I placed it in a numbered list. This process appears to be putting the namable number into a one to one correspondence with the integers:
My list has three columns. In the first column is a list of integers, in the second column is a list of named numbers, in the final column is the decimal representation of the number. I will use the third column in a moment.
(I ask my humble audience not to point out the obvious fact that the second column cannot begin with one half and contain references to all the integers in the first column. Mathematics is not built on obvious observations, but on subtle nuance. If math were built on the obvious, why then, anybody on the street could do it!)
Anyway, since I have restricted my list of namable numbers to numbers that I could describe with a finite string, and the process of naming is a sequential activity, I feel very comfortable claiming that the set of "namable numbers" is as denumerable as the rational numbers.
Georg Cantor used a thing call the Diagonal Method to show that the Real numbers are not denumerable. The diagonal method is a simple device he employed to create a number not yet in a given list of real numbers.
Here is my problem: What happens if I perform the diagonal method on the set of namable numbers? "The diagonal product of the namable numbers" is a namable number. I am now left with a horrific quandary. The diagonal method just produced a namable number not yet in my list!
The string: "The diagonal product of the namable numbers" is finite. This string refers specifically to performing the diagonal method on my decimal expansion of the namable numbers.
I now have to face a number of hard choices: Either the set of namable numbers is not denumerable (after all it fails the diagonal method.) This choice is not desirable because the ASCII representation the namable numbers is a subset of the Integers.
I could try to claim that the diagonal does not produce a namable number. This is not desirable since it raises doubts as to whether or not the diagonal method produces a real. Note, the namable number challenge doesn't just affect the diagonal method, it seems to call into question any method that tries to produce a namable number not in a enumerated list of the reals.
The final choice is to face the terrible possibility that infinite sets may not be the same size as a subset of itself.
This namable number challenge is one of the many paradoxes facing transfinite theory.
©2002 Kevin Delaney