Rathjen - Heyting lezing

136 years and still going strong(?): Cantor's continuum problem

Michael Rathjen

Cantor's continuum hypothesis, CH, asserts that an infinite set of reals is either countable or of the same size as the entire continuum (so nothing exists in between). The quest for its status became enshrined in Hilbert's famous 1900 list, earning it a first place.

Hilbert announced a proof of CH in his 1925 talk «On the Infinite» which arguably inspired Gödel to find a model of set theory that validates CH. With the work of Cohen in the 1960s, many people were convinced that CH was now a settled problem as all essential facts about it were deeply understood. This situation, however, did not convince everybody (e.g. Gödel). The last 20 years or so saw a surprising resurgence of the problem in modern set theory. In the first part of the talk, I plan to survey some important known facts about CH. This will be followed by relating some of the modern attempts at finding new axioms to settle CH. Their justifications are partly based on technical results but also tend to be highly speculative and often come enmeshed with essentially philosophical arguments.

Finally I shall consider a specific proposal by Solomon Feferman to clarify the nature of CH. He has advocated an account of the set-theoretic universe which allows him to distinguish between definite and indefinite mathematical problems. His way of formally regimenting this informal distinction is by employing Heyting's logic for domains for which one is a potentialist and reserving classical logic for domains for which one is an actualist. This framework allowed him to state a precise conjecture about CH, which has now been proved. I plan to indicate a rough sketch of the proof which features realizability and forcing.