four postulates]." Fermat's Last Theorem Fermat's Last Theorem was conjectured by Pierre de Fermat in the 1600s, states the impossibility of finding solutions in positive integers for the equation with . Fermat himself gave a proof for the n = 4 case using his technique of infinite descent, and other special cases were subsequently proved, but the general case was not proven until 1994 by Andrew Wiles. Richard's paradox This profound paradox presented by Jules Richard in 1905 informed the work of Kurt Gödel and Alan Turing. A succinct definition is found in Principia Mathematica: Kurt Gödel considered his proof to be “an analogy” of Richard's paradox, which he called “Richard's antinomy” . Alan Turing constructed this paradox with a machine and proved that this machine could not answer a simple question: will this machine be able to determine if any machine (including itself) will become trapped in an unproductive ‘infinite loop’ (i.e. it fails to continue its computation of the diagonal number). Incompleteness of axiomatic systems: Gödel's proof To quote Nagel and Newman (p. 68), "Gödel's paper is difficult. Forty-six preliminary definitions, together with several important preliminary theorems, must be mastered before the main results are reached". In fact, Nagel and Newman required a 67-page introduction to their exposition of the proof. But if the reader feels strong enough to tackle the paper, Martin Davis observes that "This remarkable paper is not only an intellectual landmark but is written with a clarity and vigor that makes it a pleasure to read" (Davis in Undecidable, p. 4). It is recommended that most readers see Nagel and Newman first. Gödel proved, in his own words: "It is reasonable... to make the conjecture that ...[the] axioms [from Principia Mathematica and Peano ] are ... sufficient to decide all mathematical questions which can be formally expressed in the given systems. In what follows it will be shown that this is not the case, but rather that ... there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis of the axioms" (Gödel in Undecidable, p. 4). Gödel compared his proof to "Richard's antinomy" (an "antinomy" is a contradiction or a paradox; for more see Richard's paradox): "The analogy of this result with Richard's antinomy is immediately evident; there is also a close relationship [14] with the Liar Paradox (Gödel's footnote 14: Every epistemological antinomy can