Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non professionals.

Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.

• This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.

Jaakko Hintikka 1. How to Study Set Theory The continuum hypothesis (CH) is crucial in the core area of set theory, viz. in the theory of the hierarchies of infinite cardinal and infinite ordinal numbers. It is crucial in that it would, if true, help to relate the two hierarchies to each other. It says that the second infinite cardinal number, which is known to be the cardinality of the first uncountable ordinal, equals the cardinality 2 o of the continuum. (...) (Here o is the smallest infinite cardinal.). (shrink)

In a recent article (‘The Continuum: Russell’s Moment of Candour’), Christopher Ormell argues against the traditional math- ematical view that the real numbers form an uncountably infinite set.1 He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of (...) study. In practice, this means excluding as inadmis- sible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. (shrink)