Ineffable limits of weakly compact cardinals and similar results

It is proved that if an uncountable cardinal κ has an ineffable subset of weakly compact cardinals, then κ is a weakly compact cardinal, and if κ has an ineffable subset of Ramsey (Rowbottom, Jónsson, ineffable or subtle) cardinals, then κ is a Ramsey (Rowbottom, J\'onsson, ineffable or subtle) cardinal.

More on bases of uncountable free abelian groups

We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.

More on bases of uncountable free abelian groups

We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.

Knaster and friends II: The C-sequence number

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [Formula: see text] and independence results about the [Formula: see text]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [Formula: see text]-sequence spectrum and uncover some tight connections between the [Formula: see text]-sequence spectrum and the strong coloring principle [Formula: see text], introduced in Part I of this series.

A version of $$\kappa $$-Miller forcing

We consider a version of $$\kappa $$ κ -Miller forcing on an uncountable cardinal $$\kappa $$ κ . We show that under $$2^{<\kappa } = \kappa $$ 2 < κ = κ this forcing collapses $$2^\kappa $$ 2 κ to $$\omega $$ ω and adds a $$\kappa $$ κ -Cohen real. The same holds under the weaker assumptions that $${{\,\mathrm{cf}\,}}(\kappa ) > \omega $$ cf ( κ ) > ω , $$2^{2^{<\kappa }}= 2^\kappa $$ 2 2 < κ = 2 κ , and forcing with $$([\kappa ]^\kappa , \subseteq )$$ ( [ κ ] κ , ⊆ ) collapses $$2^\kappa $$ 2 κ to $$\omega $$ ω .

AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN

We show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.

A Forcing Axiom Deciding the Generalized Souslin Hypothesis

We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal$\unicode[STIX]{x1D706}$, if$\unicode[STIX]{x1D706}^{++}$is not a Mahlo cardinal in Gödel’s constructible universe, then$2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$entails the existence of a$\unicode[STIX]{x1D706}^{+}$-complete$\unicode[STIX]{x1D706}^{++}$-Souslin tree.

UNIVERSAL CLASSES NEAR ${\aleph _1}$

Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

THE COMPLEXITY OF THE EMBEDDABILITY RELATION BETWEEN TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE

We prove that for every uncountable cardinal κ such that κ<κ = κ, the quasi-order of embeddability on the κ-space of κ-sized graphs Borel reduces to the embeddability on the κ-space of κ-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability relation on the κ-space of κ-sized R-modules, for every $\mathbb{S}$-cotorsion-free ring R of cardinality less than the continuum. As a consequence we get that all the previous are complete $\Sigma _1^1$ quasi-orders.