An Examination of the Philosophical Significance of Cantor's Diagonal Argument

Most contemporary mathematicians have accepted Cantor's transfinite set theory. In this theory, Cantor's diagonal argument holds special significance and has many interesting and unusual applications. To understand the nature of the diagonal argument and to assess its value and credibility, it's better to employ the diagonal argument in a problem that involves fewer but more self-evident concepts and preliminaries. Therefore, in this article, we have used the diagonal argument to prove a philosophical problem. The philosophical nature of the problem under study has led the presented diagonal argument to be based solely on concepts such as "object", "order", and "actual infinity" without the need for the more complex mathematical concepts presented by Cantor's diagonal argument. It becomes clear that the philosophical value and significance of the presented diagonal argument is not any less than proofs like the proof of correspondence (taṭbīq), the proof of endpoint and midpoint (ṭaraf wa wasaṭ), the proof of ladder (sullamī), the proof of convergence (musāmatah) and other similar proofs.