Father of Set Theory (and Infinity Infinities) Part 2

Last time we talked about how Cantor began exploring infinities by saying they could behave like integers in the fact that they could be added together and that one infinity could be larger than another infinity. He applied rules of a one-to-one basis on numbers to show that the sets of infinity were the same sizes of the infinite sets in the fact they had the same number of elements. Cantor realized that this could not be applied to decimal numbers which included irrational numbers. He had to think of several clever arguments in order to prove this, one of them being the “diagonal argument.” This argument proved that it was always possible to construct a new decimal number that was missing from the original list and thus makes decimal numbers infinite in nature, even more infinite than natural numbers. The diagonal argument is as follows and corresponds to the picture above:

Imagine an infinite set of numbers made up of an infinite pattern of just two digits (the example will use m and w). A new number can always be created by making sure that the first digit of the new number is different from the first digit of the first number in the set, the second digit is different from the second number and so on and so forth. This is the “diagonal argument” because the the number (depicted as blue in the graphic) is different in every place from the diagonal digits (which is red in the picture). By this argument, a new number could never be a duplicate of any number in the infinitely long set. Through this, an infinite set of numbers cannot contain all possible numbers which shows that there are more sets of numbers than there are numbers.

The diagonal argument went on to show that decimal numbers were “non-denumerable” or “uncountable” meaning that the set contained more elements that could ever be counted. Cantor also argued that there are in infinite number of irrational numbers in between every rational number. This shows patternless decimals filling in the “spaces” between patterned rational numbers. Cantor also coined the new term “transfinite” in order to distinguish between various levels absolute infinity. Absolute infinity was tied to God (since Cantor was a religious man and saw no contradiction between his work in mathematics and God). Since the cardinality (size) of a finite set is a natural number indicating the number of elements in a set, he needed a symbol or idea to express the size of a infinite set. Cantor resorted to using the Hebrew letter aleph (). He defined aleph-not (0) as the cardinality of the countably infinite set of natural numbers. Aleph-one ( 1) as the next larger cardinality, that of the uncountable set of ordinal numbers and so on and so forth. Because of this distinction and the properties of sets, Cantor was able to show that 0 + 0 = 0 as well as  0 x 0 = 0.

Through all of his work, Cantor provided a revolutionary step to mathematical thought which also provided the possibility of more infinities. For example an infinity (or even many infinities) between the infinity of the whole numbers and the larger infinity of the decimal numbers (henceforth the title of infinity infinities). This was known as the continuum hypothesis and Cantor believed, but could no show, that there was no such intermediate infinite set. This remained unproven until the 1960s-until the work of Paul Cohen. The next blog post will explore how he defined and refined set theory to what we know it as of today as well as some pushback from other mathematicians that Cantor received for his earth-shattering work.


Edits: Some edits were made for coherence and historical accuracy.

Sources: Mastin, L. (2010) 19th Century Mathematics-Cantor. Retrieved from: https://www.storyofmathematics.com/19th_cantor.html