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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 …

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 …

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, …

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 …

Turing's work on computability, extending those of Goedel and the other early logicians, paved the way for the development of modern computers. Before Turing and Goedel, the concept of computability w …

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 …

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 …

The subject known for decades as recursion theory, studying the class of recursive functions and the recursively enumerable (r.e.) sets and degrees, is now known almost universally, especially amongst …

The large cardinal hierarchy in set theory can be seen as an example of the phenomenon. There seems to be little reason initially to have expected that questions about what kinds of infinite sets exis …

There is a general pattern of inquiry in mathematics and the sciences by which an investigation begins in philosophy, using philosophical ideas that may be initially quite vague, but which become incr …

Appeals-to-authority, of course, are widely regarded in many contexts as logical flaws, as something to be avoided in logical discourse. But I believe that there is a sense, nevertheless, in which app …

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 …

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 …

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 …