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Since your algebra mixes together the objects $x$ and $y$ with their pair $(x,y)$ in the same algebra, it has the effect of erasing "ordered-pair" as a separate type in this context, and so there is n …

The Categories will be fundamentally different. The category of groups with group homomorphisms, (even with monomorphisms) enjoys a directedness property: any two groups can map monomorphically into t …

I'm not sure if this is what you want, but it is not difficult to prove that if f and g are primitive recursive functions, then the set A = { (x,y) | f(x) = g(y) } is a primitive recursive subset of t …

The Lindenbaum algebra is a natural Boolean algebra associated with any theory $T$. The Lindenbaum algebra can be taken to consist of equivalence classes of formulas, where two formula are equivalent …

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 …

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 …

Some of you may want simply to learn set theory, rather than learn set theory in order to do category theory. Therefore, I list here a few of the most canonical texts used by set theorists---these boo …

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 …

The answer is yes. One can do this with pointed directed graphs. Specifically, for any infinite cardinal $\lambda$, let $C_\lambda$ consist of all structures of the form $\langle V_{\lambda+2},{\in}, …

For those readers unfamiliar with the class of primitive recursive functions, let me say that you may simplify things somewhat by fruitfully thinking of them as poor cousins of the computable function …

Adrian Mathias has written some excellent articles comparing the specific set theory used by Mac Lane and used in other parts of category theory. His article The strength of Mac Lane set theory is a …

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 …

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 …

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 …