Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

60 the mathematical infinitysimple, but with a solution that no one, including Archimedes, could write down, forthe purpose of unmasking his false competitors?Archimedes’ evaluation of the area of the sphere, his interest in extremely largenumbers, and his use of passage to the limit all indicate his acceptance of infinityin mathematics. Instead, in Aristotle’s view, infinity is only something potential, butcannot ever be actually reached. Thus, mathematicians do not need an actual infinity,rather only finite quantities, as well as constructions repeated as many times as thecase may be. So, should we view Archimedes’ evaluation of the area of the sphere asa theorem or only a truth that transcends the realm of mathematics, because achievedonly through an infinite limiting process? Should we reject any result outright, onthe ground that it is obtained by forbidden methods? Forbidden by whom, and why?Because of tradition? Because of the fear of challenging accepted wisdom?2.4 Combinatorics and InfinityToday, the matter remains the subject of hot debate among mathematicians. For example,are computer proofs acceptable? And if so, under which limitations? Rejectinginfinity also leads to intrinsic difficulties. For example, as briefly discussed later, afterthe work of Paris and Harrington (1977) we know that there are simple, plausiblestatements in finite combinatorics that are provably true within the model of Zermelo-Fraenkel set theory, but are undecidable within the very natural, but much simpler,finitistic model P of Peano arithmetic. 5The combinatorial statements alluded to are a profound extension of the elementary(but not so easy to prove) “friends and strangers” theorem: Let n be given. Then inany party of sufficiently large size (determined by n) we can find either a subset ofn people who know each other (i.e., a clique) or a subset of n people who never metbefore (i.e., an independent set). How big should the size of the party be? If we denoteby R(n, n) the minimum size allowable, it is very easy to show that R(3, 3) = 6, lesseasy that R(4, 4) = 18, and beyond this we know only some inequalities, for example,43 ≤ R(5, 5) ≤ 49 and 102 ≤ R(6, 6) ≤ 165 (Fig. 2.2).Even the simplest Ramsey numbers, the numbers R(n, n), are extraordinarily difficultto determine, because the number of possible relations among n people growsincredibly fast with n. The mathematician Paul Erdős described the situation in thisway: “Imagine an alien force, vastly more powerful than us, landing on Earth anddemanding the value of R(5, 5), or they will destroy our planet. In that case, we shouldmarshal all our computers and all our mathematicians, and attempt to find the value.But suppose, instead, that they asked for R(6, 6), we should attempt to destroy thealiens.”5 See Paris and Harrington (1977). From Mathematical Reviews, MR0491063 (58 #10343): “. . . D8. J. Parisand L. Harrington: A mathematical incompleteness in Peano arithmetic. This research paper is included herebecause of its importance. An arithmetically expressible true statement from finitary combinatorics is givenwhich is not provable in Peano arithmetic P. The statement S in question is the strengthening of the finiteRamsey theorem by requiring the homogeneous set H to be ‘relatively large’, i.e., |H |≥min H . The statementS follows from the infinite Ramsey theorem and is equivalent in P to the uniform ∑ 01 -reflection principle for P.The corresponding Ramsey function exceeds almost everywhere all provably recursive functions of P.”