Cantor and Me

Britannica.com
Georg Cantor, (born March 3, 1845, St. Petersburg, Russia—died Jan. 6, 1918, Halle, Ger.), German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another.

I first encountered Cantor in a book I read while in Jr. High. I wasn't quite sure what to make of his notion of transfinite numbers, so I noted it and went on with my life.

I encountered him again in High school, junior year, in geometry class. The author of the book (Geometry with Coordinates) wrote that a square contains an infinite number of points. So does a line segment bordering the square. But the impossibility of making a one-to-one correspondence between the two infinities is another example of different degrees of infinity.

I didn't think so, but was not able to prove my doubt until around 30 years ago ...

Any point in a square can be identified by a pair of coordinates (X,Y), where each is a number from 0 to 1.

X can be expressed in the form: 0.abcde...

Likewise, Y can can be expressed as 0.ABCDE...

It is then possible to generate another number, Z, by interdigitating the values for X and Y.

Z = 0.aAbBcCdCeE...

Z represents a unique point in a line segment, and the mapping is reciprocal because the value of Z can be decomposed into the original values of X and Y.

     .a b c d e  (X)
Z = 0.aAbBcCdDeE...
     . A B C D E (Y)

Furthermore, by taking the digits of Z in groups of three, a unique and reciprocal correspondence can be established between the points of the line segment (Z) and the points within a cube (R,S,T).

R = aBdE...
S = AcDf...
T = bCeF...

In addition, we can map a square onto a circular area. Let the radius of the circle equal 1. By dividing the angles by 360, we get values ranging from 0 to 1. In this way any point within the circle is uniquely mapped to a point within the square. Or to a line segment. Or to a cube.

So much for the textbook author.

As for Cantor, there has been much confusion about what he actually wrote. And when he wrote it. I don't know that Cantor was a follower of Hegel, but he did live during the time of what Nietzsche called the "Hegelian Infection", with its talk of infinities and intimations.

I remember from school (I don't recall which year) that Real numbers are somehow different from integers, to the extent that Real 2 and Integer 2 are different numbers. Apparently someone mistakenly supposed that this was how the infinite set of Real numbers could be larger than the infinite set of Integers (or Natural numbers).

In fact, Cantor actually wrote of the numbers of the "continuum", meaning the numbers between 0 and 1. This is a very different matter.

And of course it does appear impossible to enumerate the numbers of the continuum because, wherever we start, we run into another infinity. That is, between .5 and .6 there is an infinite number of points. And between .5 and .55 there is an infinity. Also between .5 and .51. Also, Cantor proved that the set of "irrational" numbers is infinite. And because (he claimed) it is not possible ennumerate all the rational and irrational numbers, the set of continuum numbers must be infinitely greater than the infinite set of natural numbers.

What to do?

While working on a minor programming problem, where I had to sort some non-numerical values, I had an idea.

The items were designated R1, R2, R3, etc. A simple ASCII sort arranged them as R1, R10, R11, R12 ... R19, R2, R20 ... This was not what I wanted.

It occurred to me that these items could be brought into the desired arrangement by doing a dual sort; i.e., sorting by string length +ASCII value. This gave me R1 thru R9 (string length = 2) followed by R10 thru R99, etc.

I applied this little insight to the problem of mapping numbers of the continuum to natural numbers. I would start by sorting (in my imagination) the continuum numbers according to string length (number of significant digits), and then by value.

This resulted in Mirror Mapping, where a natural number was mapped to its reversed image.

That is, 1 maps to .1; 10 maps to .01; and 12345 is mapped to .54321.

In this way, the natural numbers are mapped to the 1 digit C-numbers, then the 2 digit, and the 3 digit and so on forever and ever.

Cantorists will complain that this mapping provides no unique natural number for an irrational number. That is true, but irrelevant because the mapping provides an infinite series of approximations for each irrational number (e.g., 1 digit, 2 digit, 3 digit, ad infinitum).

Does this mean that the set of Natural numbers is infinitely larger than the infinite set of Continuum numbers?

Not at all. Infinities are slippery. When it comes to mapping, we should not mistake our lack of ingenuity for actual impossibility.

PS,

A Prime Proof

While I was entertaining myself mapping infinities, I found myself thinking, "If I knew the set of prime numbers was infinite I could ..."

And before I could finish the thought, the proof fell out of my head and onto the paper. It had bugged me for years. Ever since my 9th grade science teacher told me that there was some question as to whether the set of prime numbers was infinite. I had assumed that it was, but hadn't seen a proof.

At that time, and for years after, everytime I thought about the set of prime numbers I found myself pondering some variation on the Sieve of Eratosthenes. Lots of variations, but no proof. Not until I had worked out the infinities business.

The proof is embarassingly trivial, once I got it written down. I already knew that if a proposition cannot be proven directly, it may useful trying to prove the opposite. But I didn't apply that rule. At least not consciously.

P! + 1

That does need a little explanation. If we assume that the set of prime numbers is finite, there must be a largest prime number. Call it P.

And if P is the largest prime number, then P! [P factorial] contains all the prime numbers as roots.

1) Every natural numbers is either a prime or products of primes
2) Consecutive natural numbers cannot share any prime root other than 1.

Therefore, P!+1 is either a prime number (larger than P), or a product of primes larger than P.

P is not the largest prime number.

And because the process can be repeated without limit, e.g., taking P!+1 as the presumed "largest prime ", there can be no largest prime number.

Therefore, the set of prime numbers is infinite.

QED

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