The treatment of infinity in mathematics has separated mathematicians into different schools. Cantor’s discoveries opened the 20th century debate about the meaning of numbers, and Cantor introduced the idea that there may be an actual infinite. mathematics begins to sound like theology.
Cantor’s diagonal argument shows that there is not only countable infinity, but that the real numbers are uncountable. This creates different types of infinity, and Cantor introduced a new type of number, aleph, to try to build a way to calculate infinity. The following little introduction discusses these types of numbers in mathematics.
Here is a quote from Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 216
“As a result of Cantor’s developments, one could divide the mathematical community into three sorts. There were the finitists, typified by the attitudes of Aristotle or Gauss, who would only speak of potential infinities, not of actual infinities. Then there were the intuitionists like Kronecker and Brouwer who denied that there was any meaningful content to the notion of quantities that are anything but finite. Infinities are just potentialities that can never be actually realised. To manipulate them and include them within the realm of mathematics would be like letting wolves into the sheepfold. Then there were the transfinitists like Cantor himself, who ascribe the same degree of reality to actual completed infinities as they did to finite quantities. In between, there existed a breed of manipulative transfinitists, typified by Hilbert, who felt no compunction or need to ascribe any ontological status to infinities but admitted them as useful ingredients of mathematical formalism whose presence was useful in simplifying and unifying other mathematical theories. “No one,” he predicted, “though he speak with the tongue of angels, will keep people from using the principle of the excluded middle.“
The following video is a nice introduction to the problem of infinity in mathematics and reality. It is by Paul Budnik. Here is his introduction:
“There are two approaches to mathematical infinity. It can be seen as defining limiting cases that can never be realized or as existing in some philosophical sense. These mathematical approaches parallel approaches to meaning and value that I call absolutist and evolutionary. The absolutist sees ultimate meaning as something that exists most commonly in the form of an all-powerful infinite God. The evolutionary sees life and all of creation as an ever-expanding journey with no ultimate or final goal. There is only the journey. There is no destination. This video argues for an evolutionary view in our sense of meaning and values and in our mathematical understanding. There is a deep connection between the two with profound implications for the evolution of consciousness and human destiny.”