At least as far as first-order theories go, one could construe the (downward) Löwenheim–Skolem theorem as an incentive to invest more in discrete models rather than in continuous ones. This would ...
As probably many other people here, I learned integration, as an undergrad, from Rudin's books. I recently realized, however, that I don't quite use Lebesgue integration in my work, or at least I use ...
How would you explain one of these theorems in the foundations of mathematics to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the ...
I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...
Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...
This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...
Of all the constructions of the reals, the construction of the surreals seems the most elegant to me. It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...
In the theory of programming languages and structural proof theory, one of the handiest techniques we have available is a method called "logical relations", in which you can prove properties of ...
Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in ...
When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the ...