Australian Senior Mathematics Journal vol. 25 no. 1

“reasoning involving proof—the one compelling argument for teaching mathematics” in the new national draft curriculum (ACARA, 2010). The rationale for the subject Queensland Senior Mathematics B (Queensland Studies Authority, 2008) includes the aim that students should appreciate the “nature of proof” and “the contribution of mathematics to human culture and progress”. Although the study of proofs is not specifically recommended in it, the draft curriculum (ACARA, 2010) advocates as an aim the development of “thinking skills” and “creativity” in students. The study of Kurt Gödel’s proof of the “incompleteness” of a formal system such as Principia Mathematica (Russell & Whitehead, 1910–1913) is a great way to stimulate students’ thinking and creative processes and interest in mathematics and its important developments. This paper describes salient features of the proof together with ways to deal with potential difficulties for students. It recommends the study of the logical-skeletal structure before students attempt the proof itself. It describes how students can be introduced to the proof with a documentary highlighting its importance (Malone & Tanner, 2008); two books for the ‘general reader’, Nagel and Newman (2001) and Frantzen (2005) are evaluated and the best description of its logical core written in clear English (Feferman, 2006c) is given. The author also suggests a prior discussion about paradoxes in mathematics with students, in particular the Richard paradox, the Liar’s paradox—“This sentence is false,”—and Russell’s set-theoretical paradox in the theory of classes (Hersh, 1998). Bertrand Russell and A. N. Whitehead’s Principia Mathematica (1910–1913), hereafter designated as PM, contained a proof that the whole of mathematics can be developed on the basis of set theory. With it they hoped to prove that all mathematics is founded on logic. Kurt Gödel’s proof (1931/1986) of the ‘incompleteness’ of formal systems such as PM is important for many reasons. It is important in the history of mathematics and for further developments in mathematics such as: the