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Theodor Nenu and I have a paper addressing the question of whether Alan Turing proved the undecidability of the halting problem in his seminal 1936 paper on computable numbers, in which he introduces …

W. Hugh Woodin, at a 1992 seminar in Berkeley at which I was present, proposed a new and ridiculously strong large cardinal concept, now called the Berkeley cardinals, and challenged the seminar audie …

For the philosophy of mathematics, I wrote my book specifically with mathematical readers in mind. Many readers have told me that they appreciate the accessible manner the book has of treating even su …

Your question is answered by the distinction between the first-order and second-order Peano axioms. The categoricity result of Dedekind refers to the second-order Peano axioms rather the first-order a …

Nearly every mathematician nowadays is familiar with the fact that there is up to isomorphism only one complete ordered field, the real numbers. Theorem. Any two complete ordered fields are isomorphic …

Dedekind actually in effect gave two different definitions of infinity. Namely, first, as is well known, a set is Dedekind infinite if it is equinumerous with a proper subset of itself. But second, De …

When I was a graduate student at Berkeley in the early 90s, I had heard that Silver's approach to refuting ZFC involved the idea that somehow we make a mistake in our thinking about ZFC by conflating …

On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep. For example, one of the critical ideas in my paper Joel David Hamkins, Gap forcing, …

This is a side matter to the main question here, but I wanted to add a bit more on the history of the universe concept, since this seems to be less widely known than it deserves. Namely, universes wer …

Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning. With induction, one uses a comparativel …

Computability theory overflows with hierarchies of computability notions, which allow us precisely to compare the computational strength of diverse notions. We have the hierarchy of Turing degrees, wh …

Although the axiom of constructibility is often resisted by set theorists with the view that it is restrictive, nevertheless there are a variety of ways in which the axiom is compatible with strength …

Adrian Mathias has written a number of excellent essays criticising various aspects of Bourbaki's logical foundations, and I encourage you to follow the link and read them. He writes supremely well, a …

Here is an example from set theory. Set theorists commonly study not only the theory $\newcommand\ZFC{\text{ZFC}}\ZFC$ and its models, but also various fragments of this theory, such as the theory o …

Allow me to mention that since the players in effect adopt the roles of the quantifiers $\forall$ and $\exists$, as Bob has a winning strategy just in case for every move for Alice, there is a reply b …