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

combinatorics and infinity 61Figure 2.2. A proof that R(3, 3) ≥ 6. The vertices represent five people: those connected by anouter edge have met each other, and those connected by an inner edge never met before. Thiscomplete graph on five vertices has no triangle with edges of a same type.The situation for the Paris-Harrington result is even worse. An apt description of thisdifficulty is that the problem is like the Hydra monster of Greek mythology, which grewback two heads when one was cut off (Fig. 2.3). Solving the Paris-Harrington riddle inPeano arithmetic is like killing the Hydra, because the examination of each case givesrise to several more subcases, and so on ad infinitum. A solution can be found only byan excursion all the way to infinity, where we can see what happens, and only then,bringing the knowledge so obtained back to the finite realm, can we get the answer.More precisely, the Paris-Harrington problem arises as follows. Consider arbitrarycolorings, with c colors, of all subsets of {1, 2,...,r} and ask for a subset E, ofcardinality n, such that all subsets of E with a same cardinality have the same color.The fact that we can always do this, provided that r is sufficiently large as a functionof n and c, is a theorem in P. The Paris-Harrington problem consists in adding theapparently mild restriction on E that its minimum element does not exceed n. The factthat we can always find such an E if r is sufficiently large remains true, but thecorresponding function grows so incredibly fast that it is not computable in the finitisticmodel of P. The statement is, however, provable in the arithmetic of P after adding toit the transfinite induction up to the ordinal ɛ 0 .The first and simplest approach to the mathematical infinity is the counting withoutend. This naive definition is tautological because, as pointed out by Georg Cantor,infinity is then implicitly defined self-referentially as a process that goes on for infinitetime. Mathematicians avoid such a paradox by axiomatizing the rules of mathematicsin a small set of axioms, thus reducing any mathematical proof to a logical combinationof such axioms. The pitfall of this approach is that, because of Gödel’s incompletenesstheorems, we cannot prove that the axioms themselves are not contradictory and neverlead to statements that are true and false at the same time. At any rate, because noprofound inner contradiction has been found in the body of all of mathematics doneup to today, mathematicians tend to ignore the question and go on happily with theirwork, leaving the subtleties of foundational mathematics to historians, philosophers,and logicians. Here I will do the same and ignore foundational questions.