Monday, December 10, 2018

Infinity contra-Cantor #1

In the late 19th century Georg Cantor chose the method of 1-1 correspondence to decide the different quantities of infinite sets. He called such quantities their “cardinality.” If the members of one set can be paired 1-1 with the members of another set, then they have the same “cardinality” (link). 

As the Wikipedia page shows, some famous mathematicians such as Kronecker and PoincarĂ© objected to some of Cantor’s ideas. But most people who have an opinion about it accept Cantor’s arguments and consider that the cardinality of the positive integers equals that of the even positive integers, as explained on the linked page.

I was astounded when I became aware of that many years ago. Years later I realized Cantor's error. The assumed method of 1-1 correspondence implicitly rules out 2-1 correspondence, N-1 correspondence, and part-whole logic

A 2-1 correspondence or mapping can be formed from the natural numbers (integers) to the even numbers as follows: 1, 2 → 2;   3, 4 → 4;   5, 6 → 6;   7, 8 → 8;   9, 10 → 10; and so on. If this mapping did not include “and so on”, then the finite quantity of natural numbers would surely be regarded as being twice that of the even numbers. (See above link, Exercise 3). However, many abandon that idea when “and so on” is included. I do not. From a part-whole logic perspective, imagine having a container with all the integers in it, then removing the odd integers from the container. Wouldn’t the container then have half the quantity of numbers as before? Or partition the container in half vertically, putting the odd integers on one side of the partition and the even integers on the other side. Wouldn’t there be twice as many numbers in the whole container as there are on one side of the partition?

1 comment:

  1. The 1-1 correspondence rule works for finite collection. By 1-1 matching one can tell whether the first collection has fewer/just as many/more elements as the second collection. Cantor simply extended this method of comparison to non-finite sets...