Alchemist: I am very uncomfortable about what are called transfinite numbers, due to the counterintuitive nature of the subject. A man named Cantor formalized the subject prior to 1900.

Cardinality is a fancy term for how many members a set has. Cardinal numbers specify how many. Ordinal numbers specify position in a sequence. There are fifty people in line at the ticket office. I am the fifth in the line. Fifty is cardinal, while fifth is ordinal.

Cantor analyzed infinite sets and coined the term transfinite numbers. He started by pointing out that the cardinality of sets could be determined by trying to match members.

If the members of two sets can be put into one-to-one correspondence, they have the same number of members or the same cardinality. If the matching process results in one set having some unmatched members, it has more members and the greater cardinality. Very simple and intuitively easy so far.

Then he pointed out that you could match all the even integers with all the integers. There are as many even integers as there are total integers. Try it. Start writing a list of all integers. In a parallel list write a list of all the even integers. You get the following pairs: (1,2), (2,4), (3,6), (4,8) . . . For every even integer, there is a corresponding integer in the other list, and vice versa.

He used this property as the fundamental definition of a set with a transfinite number of members: Such a set could be put into a one-to-one correspondence with a subset of itself. He went on to prove some interesting theorems, including the following.
  • The set of all rational numbers has the same cardinality as the set of all integers. There is a systematic way of matching the members of these two sets.
  • There are more real numbers than integers. The number of real numbers is called the Power of the Continuum. It is the number of points on a line.
  • The number of points on a short line has the same cardinality as the number of points on an infinitely long line.
  • The number of points in a 2D or 3D space is the same as the number of points on a line, or 1D space.
  • If you make a set of all the subsets of a set, it has a larger cardinality than the original set.
All very counterintuitive. He called the cardinality of the set of all integers Aleph-0 (using the first letter of the Hebrew alphabet). Of course this is the same as the cardinality of various other sets. The set of all subsets of the set of integers he said had cardinality Aleph-1. Similarly, you can define Aleph-2, Aleph-3, et cetera.

I think he proved that the Aleph’s were all the transfinite sets there could be, by proving that there could be no transfinite number between Aleph-n & Aleph-(n+1).

I think there is an unproven conjecture about the Power of the Continuum being Aleph-1.

Does the above help or confuse you? Most of it confuses me. I agreed with all the logic when I read about transfinite sets, but find the conclusions counterintuitive. Emotionally, I think there are more rational numbers than integers, but intellectually I agree with the proof that both sets have the same cardinality.