In this paper we complete the characterization of Ext(G,Z) under Godel's axiom of constructibility for any torsion-free abelian group G . In particular, we prove in (V=L) that, for a singular cardinal nu of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence of cardinals (nu_p : p in P) satisfying nu_p <= 2^{nu}, there is a torsion-free abelian group G of size nu such that nu_p equals the p-rank of Ext(G,Z) for every prime p and 2^{nu} is the torsion-free rank of Ext(G,Z).
We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.
Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $λ$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $κ$ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.
Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $λ,$ there is no universal graph of size $λ$.
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_ω$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{ω+1}$ Borel sets.
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and fin…
Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $κ$, there are $κ^+$-Aronszajn trees and all such trees are special.
Assuming the existence of a certain hod pair with a Woodin cardinal that is a limit of Woodin cardinals, we show that the Chang model satisfies $\mathsf{AD}^+$ in any set generic extensions.
Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $κ$, there are no $Σ^1_1(κ)-κ-$mad families.
Assuming the existence of certain large cardinal numbers, we prove that for every projective filter $\mathscr F$ over the set of natural numbers, $\mathscr{F}$-bases in Banach spaces have continuous coordinate functionals. In particular, this applies to the filter of statistical convergence, thereby we solve a problem by V. Kadets (at least under the presence of certain large cardinals). In this setting, we recover also a result of Kochanek who proved continuity of coordinate functionals for countably generated filters (Studia Math., 2012).
Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $μ>LS({\bf K})$ be a regular stability cardinal and let $χ$ be the local character of $μ$-nonsplitting. The following holds: 1. When $μ$-nonforking is restricted to $(μ,\geqχ)$-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension and continuity. It also has local character $χ$. This generalizes Vasey's result which assumed $μ$-superstability to obtain same properties bu…
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
In 1990, Comfort asked: is there, for every cardinal number $α\leq 2^{\mathfrak{c}}$, a topological group $G$ such that $G^γ$ is countably compact for all cardinals $γ<α$, but $G^α$ is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals $α$ is there a topological group $G$ such that $G^γ$ is countably pracompact for all cardinals $γ< α$, but $G^α$ is not countably pracompact? In this paper we construct such group in the case $α= ω$, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, and in the case $α= κ^{+…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the Löwenheim-Skolem-Tarski number (there are e…
Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_δ$ be a non-trivial $\in$-cofinal $Σ_1$-elementary embedding, where $\varepsilon,δ$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_δ$, mostly focusing on the case $\varepsilon=δ$. In particular, if $\varepsilon=δ$ and $j\in L(V_δ)$ then $δ$ has cofinality $ω$. However, assuming ZFC+I$_3$, with the appropriate $\varepsilon=δ$, one can force to get such $j\in L(V^{V[G]}_δ)$. Assuming Dependent Choice and that $δ$ has cofinality $ω$ (but not assuming $V=L(V_δ)$), and $j:V_δ\to V_δ$ is $Σ_1$-e…
Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $ω_2$-Aronszajn tree, the $ω_1$-approachability property fails, and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects.
狭义系统是由Magidor和Shelah在奇异基数的后继树性质的工作中引入的组合对象。类似于树的性质，基数$κ$满足\emph{narrow system property}，如果每个高度的窄系统$κ$都有一个共尾分支。本文研究了狭义系统性质、平方原理和强迫公理之间的联系。在大基数的假设下，证明了$\aleph_{ω+1}$满足狭义系统性质与$\square_{\aleph_ω, < \aleph_ω}$成立是相容的，并且每个正则基数满足狭义系统性质也是相容的.我们介绍了自然加强经典平方原则，并展示了它们如何可以用来产生狭窄的系统，没有共尾分支。最后，我们表明，适当的强制公理意味着，每一个窄系统的可数宽度有一个共尾分支，但与存在一个窄系统的宽度$ω_1$没有共尾分支是一致的。
Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic $\mathbf{GL}$ is complete with respect to the filter sequence of normal measures. This result answers a question of Andreas Blass from 1990 and a related question of Beklemishev and Joosten.
We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q$, $\langle L[P],\in ,P \rangle$ and $\langle L[Q],\in ,Q \rangle$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. The theory of such models is thus invariant under set forcing. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. One outcome is that we can chara…
