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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 …

Nelson claimed to have succeeded just now. http://www.math.princeton.edu/~nelson/papers/outline.pdf I hope consensus about this forms soon, so I can know what to do with the rest of my life. If onl …

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 …

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 …

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 …

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 …

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 …

I'm not sure if this specific claim is stated explicitly anywhere, but it follows from the more general discussion about intermediate extensions in Jech, page 247. If we take any $x \in L[r]$, then th …

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 …

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it h …

Is it still open whether ZFC+GCH is consistent with the statement that there are no $\aleph_2$-Suslin trees?

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 …

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is …

A sketch of Silver's original argument appears in section 19 here: http://math.bu.edu/people/aki/e.pdf

Axiom: $0^\sharp$ exists. $0^\sharp$ is a pivotal principle in the large cardinal hierarchy, but it is actually a set of natural numbers. If it exists, it is unique.