A half-century of independence

Just over 50 years ago, Paul Cohen announced the unsolvability of Cantor’s Continuum Hypothesis on the basis of the basic principles of Set Theory, these are the ZFC axioms.

But does this mean that the question of the Continuum Hypothesis has no answer? Any "solution" must involve the adoption of new principles but which principles should one adopt? Alternatively, perhaps the correct assessment of Cohen's discovery is that the entire enterprise of the mathematical study of Infinity is ultimately doomed because the entire subject is simply a human fiction without any true foundation. In this case, any "solution" to the Continuum Hypothesis is just an arbitrary (human) choice.

Over the last few years a scenario has emerged by which with the addition of a single new principle not only can the problem of the Continuum Hypothesis be resolved, but so can all of the other problems which Cohen's method has been used to show are also unsolvable (and there have been many such problems). Moreover the extension of the basic (ZFC) principles by this new principle would be seen as a compelling option based on the fundamental intuitions on which the entire mathematical conception of Infinity is founded.

However, this scenario critically depends upon the outcome of a single conjecture. If this conjecture is false then the entire approach, which itself is the culmination of nearly 50 years of research, fails or at the very least has to be significantly revised.

Thus the mathematical study of Infinity has arguably reached a critical point and interesting times are ahead.

About the speaker

Professor W. H. Woodin holds a joint position in the Departments of Mathematics and of Philosophy at Harvard University; he was previously Professor at the University of California, Berkeley, from 1989 to 2014. In 1985 Professor Woodin was awarded the `Presidential Young Investigator Award', and he received the `Hausdorff Medal' of the European Set Theory Society in 2013. He was a plenary lecturer at the International Congress of Mathematicians in 2010, and a section lecturer in 1986 and 2002. He featured in the BBC Horizon programme `To Infinity and Beyond' in 2010.

Professor Woodin is one of the leaders in our era in the quest to understand the fundamental nature of sets and the real numbers, taking forward the journey of Godel and Tarski into the far reaches of `higher cardinals'.

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