Diagonal Arguments

I develop a new account of Socratic elenchos, starting from the observation that Plato's Socrates asks not one but two logically determinative questions in elenchos: the ti esti, or, “What is —?” question, and the peri tinos, or, “— about what?” question. On this basis, I obtain the following results: The two questions interact systematically to generate a series of strict dilemmas, including those defining the problems of self-knowledge in the Charmides and of the idea of the good in the Republic. These dilemmas anticipate and are explicable by diagonalization, the key to the theorems of Cantor, Gödel, et al. They are also surprisingly convergent with the negative results of the dialectic of in-itself and for-itself in Sartre's Being and Nothingness and related works. The nexus of Platonic and Sartrean reflections on duality, suitably formalized and generalized, is the thesis that there is not, on pain of contradiction, a being which is exactly what it is of and is of exactly what it is, even though such a synthesis of being and being-of can be recognized as the structure of value, or the projection of a perfect being. Metalogical reflection having become the witness of this contradiction, a structural alteration is occasioned in value, enabling the form of the good to distinguish itself from that of the perfect. I call “negative foundations” the metaphilosophical position that results when one puts such dualities, especially as obtained via diagonalization, at the center of logical and ethical reflection. Negative foundations provides a fruitful alternative to familiar foundationalisms and antifoundationalisms.

This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly proves and the logical absurdity of "uncountable sets" that are deemed larger than the set of natural numbers. Cantor's diagonalization proof (CDP) is used to prove various things like Gödel's Incompleteness theorem (GIT) and the uncountability of real numbers (URN). In the case of GIT, it is assumed that you can have a complete list of provable statements. You use CDP to prove a statement that is not on the list, then conclude the statement is not a provable statement; otherwise, it would be part of the list of all provable statements. However, if your complete list is not really complete, then it makes sense that you can prove a new provable statement not on the list. In the case of proving the URN, you use proof by contradiction: assume the real numbers are countable and assume a complete list of them between, say, .000 ... and .111 ... The list is believed to be complete because an infinite number of counting numbers (1 to n) each allegedly have a oneto-one correspondence to each real number on the list. CDP is then used to show there is a real number that is not on the list (a contradiction). Therefore, according to the proof, there is no bijection between the real numbers and the counting numbers. Below, we will work through an example proof using CDP and demonstrate why it comes up short. Let's begin with an nXn matrix of real numbers in base 2. We go along the diagonal and flip each bit from 0 to 1 or 1 to 0 to create a new number not on the list. If we think the number is on the list, say, at row r, we can check it and see that its bit at column r doesn't match. We can repeat this exercise for all rows and see that we truly have a new real number not on the list.