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In my thesis I proved that my conjecture is false. It's not so simple-- the proof ties together chapter 2 and sections 5.1, 5.2, and 6.3.

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

The following is a theorem of Baumgartner, Hajnal, and Mate: Suppose $J$ is a normal ideal on $\omega_1$ which is nowhere $\omega_1$-dense. Then for any sequence $\langle A_\alpha : \alpha < \omega_ …

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 …

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

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 …

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 …

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 …

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 $T$ …

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.

Jech, Thomas J. Some combinatorial problems concerning uncountable cardinals. Ann. Math. Logic 5 (1972/73), 165–198. Section 5 contains the forcing for arbitrarily big Suslin algebras. See also: …

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) …

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 …