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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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

I once asked Jensen this question, when we were at a conference at Oberwolfach. I told him that I had always assumed that the diamond $\Diamond$ principle was called $\Diamond$ because it expresses …

From the introduction of Legendre's Revolution (1794): The Definition of Symmetry in Solid Geometry, Giora Hon and Bernard R. Goldstein, Archive for History of Exact Sciences, Vol. 59, No. 2 (January …

Here is another example. The theorem is: every finite partial order can be found as a suborder of the partial order of the Turing degrees. On the one hand, one can prove this by undertaking a prio …

I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable …