How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,…,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if $y<n$, then $x*(y+1)=(x*y)*(x*1)$. The algebra $(S_{n},*)$ satisfies the self-distributivity law $x*(y*z)=(x*y)*(x*z)$ if and only if $n$ is a power of $2$, and if $n$ is a power of $2$, … Read more

Characterization of L[T2n+1]L[T_{2n+1}] as a direct limit of mice

I am asking for a reference request/proof sketch for the result of Steel that characterizes L[T2n+1] as a direct limit of mice. Given that both L[T2n+1] and M2n have a Σ2n+2 well-ordering of the reals, I think the result should be that L[T2n+1] is the direct limit of iterable mice with 2n Woodin cardinals, but … Read more

Is the following product-like space a Polish space?

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prokhorov metric. For $\mu \in \mathcal{M}_1(\mathbb R)$, let $L^1(\mu)$ denote the Banach space of $\mu$-integrable functions (mod $\mu$-null). Again, for each $\mu$, $L^1(\mu)$ is a … Read more

Forcing without choice: when countable sets yield reals

One natural way to show that a forcing adds no new reals is to show that it is countable closed (EDIT: this is somewhat misleading, see Joel’s comment below). However, it turns out that this is overkill: there are forcings not adding any reals which are not countably closed, in particular, which do add a … Read more

Conditions for Isomorphism Classes of Objects in Category to Be Sets

In the field of alg. geo. that I’m studying lately, category theory language is employed but not with the highest level of precision. The lack of precision does not obstruct or obfuscate the theory much, but I am a bit curious. One of the ideas used in this discipline is the notion of isomorphism classes … Read more

Does absoluteness imply a club dichotomy?

My question is about two types of consequence of large cardinals, considered over ZFC on their own. First, we have statements of the form, “The club filter on ω1 is an ultrafilter when restricted to ‘reasonably nice’ sets of ordinals.” For Γ a class of formulas, let “CD(Γ)” denote the statement Every Γ-definable set of countable … Read more

coding a real by reflection of stationary sets

While reading a recent preprint I encountered the following theorem: THEOREM: Assume Martin’s Maximum, κ≥ω2 is regular, ⟨Sn : n∈ω⟩ is a pairwise disjoint collection of stationary subsets of κ∩cof(ω), and z⊆ω. Then there is a β<κ such that for all n∈ω: n∈z if and only if Sn∩β is stationary. The authors attribute this to the original … Read more

Status of the dense-set version of the Halpern–Läuchli theorem

The Halpern–Läuchli theorem theorem of dimension d∈ω+1 is the following strong Ramsey theoretic statement: Given k∈ω and d perfect finitely branching subtrees Ti,i<d of ω<ω, given any coloring f:Πi<dTi→k, there exists strongly embedded subtrees T′i⊂Ti and A∈[ω]ℵ0 is the common level set shared by T′i such that f↾ is constant. Some definitions: T’\subset T is … Read more

Equivalent definitions of Woodin cardinals in ZFC−/ZFC−\operatorname{ZFC}_{-}/\operatorname{ZFC}^{-}

In our background universe V – satisfying ZFC – we say that an ordinal δ is a Woodin cardinal iff it satisfies one of the following equivalent properties: For all A⊆Vδ there is a cardinal κ<δ such that for all ν<δ there is a definable elementary embedding j:V≺M, M transitive, such that crit(j)=κ, j(κ)>ν and … Read more

Is the limit of classical Laver tables connected anywhere?

Let $A_{n}=(\{1,\dots,2^{n}\},*_{n})$ be the $n$-th classical Laver table. Then $*_{n}$ is the unique operation on $\{1,\dots,2^{n}\}$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ whenever $x,y,z\in A_{n}$. Let $L_{n}:\{0,\dots,2^{n}-1\}\rightarrow\{0,\dots,2^{n}-1\}$ be the mapping that reverses the ordering of the digits in the binary expansion of the number in $\{0,\dots,2^{n}-1\}$. More explicitly, $$L_{n}(\sum_{k=0}^{n-1}2^{k}a_{k})=\sum_{k=0}^{n-1}2^{n-1-k}a_{k}$$ whenever $a_{0},\dots,a_{n-1}\in\{0,1\}$. Let $L_{n}^{\sharp}:\{1,\dots,2^{n}\}\rightarrow\{1,\dots,2^{n}\}$ be the mapping … Read more