The mathematics curricula of Australia (ACARA, 2019), Scotland, England and America all require an understanding of proof by contradiction. Specifically, proof by contradiction is included as a Geometry topic in Specialist Mathematics (Version 8.4). In Specialist Mathematics, it is expected that students construct proofs in a variety of contexts including algebraic and geometric, use examples and counter examples (ACMSMO28) and use the quantifiers 'for all' and 'there exists' (ACMSMO27). In this article the author explores historical proofs by contradiction, such as Cantor's diagonal method and its application to Turing's proof, and discusses the value of exploring such examples with advanced senior secondary students.