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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 …

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 …

To my way of thinking, the other answers are missing an important element, a necessary feature for a mathematical tool or method to be called "trick." Namely, in order to be called a "trick," a metho …