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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 …

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 …

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 …

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 …

This category does not have co-products. To see this, let $\newcommand\B{\mathbb{B}}\B$ be any atomless complete Boolean algebra with a nontrivial automorphism $\pi:\B\to\B$. For example, the forcing …

The principle is essentially asserting that $\aleph_\alpha$ exists for every ordinal $\alpha$. More precisely, it asserts that for every well-order type $\alpha$, there is a set of cardinality $\aleph …

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 …

Of course we have been investigating a wide variety of multiverse concepts, and in this sense, yes, we have not just one, but many, multiverses. But to be sure, much of this multiverse analysis has b …

It seems to me that the Berkeley cardinals can be thought of in line with your Matryoshka analogy. Namely, a cardinal $\kappa$ is Berkeley, if for every transitive set containing $\kappa$ there is an …

Following up on my remarks in the comments, allow me to answer from the perspective of model-theoretic interpretations of theories. I view interpretations of theories as providing particularly strong …

The jump inversion theorem (Friedburg 1957) shows that any Turing degree $d$ above the halting problem is the jump of another degree $d=b'$, which means that $d$ is Turing equivalent to the halting pr …

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 …