“Contra Cantor: How to Count the “Uncountably Infinite” by Erdinç Sayan, METU
Date: Friday, 12 May
Abstract: Georg Cantor’s celebrated diagonalization argument is supposed to demonstrate that the set of real numbers has a higher cardinality than the set of natural numbers. In other words, real numbers are “uncountably (or nondenumerably) infinite” whereas natural numbers are “countably (or denumerably) infinite.” I argue that the set of real numbers is, like the set of natural numbers, denumerably infinite, not nondenumerably infinite. I first discuss an attempt to directy demonstrate the denumerable infinity of real numbers between 0 and 0.999… and show that that attempt fails. I then propose an idea about an indirect way of establishing the denumerable infinity of real numbers. My method is to use tree diagrams. For convenience and simplicity of the diagrams, I employ the binary number system to represent real numbers rather than the ordinary decimal number system.
Then I take up the task of showing that, not only the real numbers between 0 and 0.999…, but the entire set of real numbers are countably infinite. Next, I argue that Cantor’s diagonal argument is fallacious and point to the exact sources of its failure. I then move on to criticize another version of the Cantorian diagonalization maneuver to prove that the power set (set of all subsets) of natural numbers is nondenumerably infinite, with cardinality N1. I also show that another one of Cantor’s contentions is erroneous. Against his claim that the power set of real numbers is nondenumerably infinite, with cardinality N2, I show that that set too is denumerably infinite. I end with some concluding remarks.