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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 …

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept. My …

My belief is that no result in algebraic geometry that does not explicitly engage the universe concept will fully require the use of universes. Indeed, I shall advance an argument that no such results …

The surreal numbers exhibit much stronger universal properties than you have mentioned, for they also exhibit very strong homogeneity and saturation properties. For example, every automorphism of a se …

Thanks for clarifying your question. The formulation that you and Dorais give seems perfectly reasonable. You have a first order language for category theory, where you can quantify over objects and m …

A Grothendieck universe is known in set theory as the set Vκ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equiv …

Automorphism groups are studied intensively in mathematics, and these groups explicitly track the difference between isomorphisms and equality. We aren't willing to say that every automorphism of a ma …

How can category theory help my research in set theory? I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at …

Let me try to answer as a set theorist, rather than as a category theorist, since I think that your question concerns at bottom a matter often considered in set theory. Namely, the essence of your qu …

Many of the most canonical early examples of categories arise as the collection of models of a fixed first order theory, with the related model-theoretic concept of homomorphism. For example, the cate …

Introduction to pluralism A version of this question lies at the heart of the ongoing dispute on pluralism in the philosophy of mathematics. Is there at bottom just one mathematical reality? Does ever …

Your question does not seemed aimed at set theorists, but let me give a set theorist's answer. I view the set/class distinction as analogous to and ultimately no more problematic really than the othe …

The main lesson is that it doesn't matter at all which particular objects you take as the numbers and what function you use as the successor function, as long as your system fulfills the right structu …

One can take some of the standard violations of CSB with other kinds of mathematical structures and transfer them to categories. For example, with linear orders, we have the two linear orders $$\langl …

Update. My new article grows out of and extends my 2010 answer to this question. The new part is the conservativity result, showing that the Vopěnka principle has the same first-order consequences as …