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The space of ultrafilters on $\omega$ has a natural semigroup structure, and ultrafilters that are idempotent in that algebra have seen applications in combinatorics on the natural numbers, for exampl …

I'll give the proof, which was told to me by Martin Zeman. Let $G \subseteq \mathrm{Col}(\omega_1,{<}\kappa)$ be generic, where $\kappa$ is Mahlo. Suppose towards contradiction that $\square_{\omega_ …

The key here is that you don't have to force beyond $\mathrm{Add}(\kappa,\kappa^+)$ in order to lift the embedding. Suppose $G \subseteq \mathbb P$ is generic and $H \subseteq \mathrm{Add}(\kappa,\kap …

I don’t know if there’s a “canonical” writeup, but I taught a master’s course a few years ago and wrote up many details of these things here. But maybe this isn’t useful if you’re looking for somethi …

Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x) …

Suppose $K$ is an ordered field, $X$ is any set, and $F$ is a filter over $X$. Let $G$ be the reduced power of $K$ by $F$. That is, we take all functions from $X$ to $K$, then take equivalence class …

There is a theorem attributed to Hahn that every ordered field $F$ containing $\mathbb R$ is a subfield of a formal power series field $\mathbb R[[X^\Gamma]]$, where $\Gamma$ is an ordered abelian gro …

Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i.e. co …

The general notion of a direct limit of a commuting system of embeddings, indexed by pairs in a directed set, has seen heavy use in set theory. It is the same notion as in category theory. I was sur …

Let $\mathbb S_\kappa$ be the standard forcing for $\square_\kappa$ by initial segments. This is $(\kappa+1)$-strategically closed. Observation: Let $T \subseteq \kappa^+$ be stationary. If $T$ …

Suppose $f$ and $g$ are functions from $\mathbb N^+$ to itself. I want to consider the function $f^g$, where $f^g(n) = f \circ \dots \circ f(n)$, where composition is done $g(n)$-many times. Note tha …

If $\kappa$ is a singular cardinal, a scale for $\kappa$ consists of an increasing sequence $\langle \kappa_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\la …

Are there any applications of the largest large cardinals to consistency results concerning, say, cardinals below $\aleph_{\aleph_\omega}$? Or perhaps to prove results in descriptive set theory? I a …

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$: (1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in V …

It is claimed that if there are two weakly compact cardinals, then there is a generic extension in which $\aleph_2$ and $\aleph_4$ have the tree property. Assuming one knows Mitchell's forcing, what …