The Continuum Hypothesis: A Mystery of Mathematics? by Matteo Viale

The continuum is arguably the most fundamental object in all of mathematics. It is the concept behind virtually all measurements. But how many real numbers are there? How many points are on a line in Euclidean space? This is one of the great mysteries of mathematics, and it can be proven to be a mystery: by the work of Cohen in 1963, the methods sufficient for ‘everyday mathematics’ are inadequate for solving this problem. Here we report recent progress on this question.

The concept of infinity has been a sub- solve CH. Gödel in 1939 was able to cardinals is the least possible. SCH is ject of speculation throughout history. In produce a model of set theory in which true in Gödel’s model, and models of set the nineteenth century, Georg Cantor CH is true. In work for which he was theory where SCH is false can be pro- produced an elegant theory in which the awarded the Fields Medal, Cohen in duced only by combining forcing tech- notion of infinity could be treated math- 1963 introduced the forcing method and niques with the use of large cardinals. In ematically: set theory. The concepts used it to produce a model of set theory my thesis I was able to generalize a investigated in set theory are the build- in which CH is false. result of Solovay in order to prove that a ing blocks on which the whole of math- ‘popular’ forcing axiom, the proper ematics can be built. In set theory the While many set theorists (including forcing axiom (PFA), implies SCH. size of an infinite set can be measured Cohen himself) are convinced that these by its cardinality. results proved that the continuum prob- There are grounds to conjecture that lem could never be settled, many others SCH is just an example of the class of Many problems regarding cardinal num- (including Gödel) believed that they problems of ‘higher complexity’ than bers that have a simple formulation are merely showed that new ‘natural CH that can be settled by forcing unsolved even after years of effort by axioms’ of set theory powerful enough axioms. The correct definition of this outstanding mathematicians. The con- to settle the continuum problem should class of problems as well as the search tinuum problem asks for a solution of be sought. In the spirit of the latter for the solutions given by forcing the continuum hypothesis (CH), and is approach, a major achievement during axioms is a topic of active research in the first in Hilbert’s celebrated list of 23 the eighties was the proof by Martin, set theory. problems. This problem has an ambiva- Steel and Woodin that there is a ‘true’ lent nature: it is comprehensible to a axiom (projective determinacy), which Matteo Viale has been awarded the 2006 vast community of scientists, but the implies that any counter-example to CH Sacks Prize. The Sacks Prize is awarded effort devoted to solving it shows that must be very complicated – essentially by the Association for Symbolic Logic even in mathematics there are decep- ‘indefinable’ in a technical sense. for the most outstanding doctoral dis- tively simple questions which may not sertation in mathematical logic. It was have an answer. More precisely, the Recent research aims at isolating ‘maxi- established to honor Professor Gerald existence of ‘solutions’ depends on the mality properties’: principles which say Sacks of MIT and Harvard for his philosophical attitude of the person that the greatest possible variety of con- unique contribution to mathematical seeking them: does every precisely ceivable mathematical objects exists. A logic, particularly as adviser to a large stated mathematical question have a ‘natural’ class of axioms that approxi- number of excellent Ph.D. students. solution? Most Platonists believe so, mate this criterion, the so-called ‘forc- and a large number of them believe that ing axioms’, have been found. These the latest research in set theory is lead- imply the axiom of projective determi- ing to a satisfactory solution of the con- nacy and the failure of CH, as well as Links: tinuum problem in particular. A skepti- many new and surprising results in gen- http://en.wikipedia.org/wiki/ cal mathematician, on the other hand, is eral topology and infinite combina- Continuum_hypothesis apt to believe that this problem will torics. Hugh Woodin has provided a http://www.ams.org/notices/200106/ never be settled. philosophical motivation for the failure fea-woodin.pdf of CH: he has solid mathematical argu- http://www.ams.org/notices/200107/ While there are many ways to formulate ments to conjecture that any ‘maximal- fea-woodin.pdf CH, we choose the following: ity principle’ able to settle all the prob- http://www.crm.es/Publications/04/ lems of mathematics of the same ‘com- pr591.pdf For every subset X of the real numbers, plexity’ as CH will imply the failure of http://www.logic.univie.ac.at/~matteo/ either there is an injective map from X CH. into the natural numbers or there is an Please contact: injective map from the real numbers Forcing axioms solve another problem Matteo Viale into X. in cardinal arithmetic of a ‘higher com- Kurt Gödel Research Center for plexity’ than CH: the Singular Cardinal Mathematical Logic Gödel and Cohen showed that the Hypothesis (SCH) asserts that the cardi- University of Vienna, Austria accepted axioms of set theory cannot nality of the power-set of strong limit E-mail: [email protected]

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