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Update. This answer does not answer the question that was asked, since Jech is using what had seemed to me as an idiosyncratic definition of hereditary countable. But upon reflection, I find his defin …

Although people often talk as though it just doesn't matter which approach you use — perhaps all universes are alike? Let me prove that in a strict sense this is not true. The nature of the mathematic …

Perhaps this isn't what you had in mind, but the Compactness Theorem of first order logic, proved by Goedel, fits your "experimental" metaphor quite well, and variations on its theme have led to some …

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 …

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 …

There is a set theoretic axiom due to Paul Corazza called the Wholeness Axiom, which is stated in the language of ZFC augmented by a single unary function symbol $j$. The axiom expresses, as a scheme, …

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 …

Since you asked about ultraproducts, and not ultrapowers, let me argue that the answer must be negative. The reason is that the category of $L$-structures under elementary embeddings is partitioned in …

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 …

Allow me to reinterpret your question as the inquiry How can abstract infinitary constructions inform us about the finite? To my mind, this is the troubling or at least surprising possibility at the …

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 …

First of all, in ZFC set theory one cannot prove all proper classes have the same size, and consequently it is not fully sensible to refer to "the size of a proper class," since they can have differen …

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 …

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 …

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 …