Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0009063

On our paper `Almost Free Splitter', a correction

Rüdiger Göbel·Saharon Shelah
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let R be a subring of Q and recall from math.LO/9910161 that an R-module G is a splitter if Ext_R(G,G)=0. We correct the statement of Main Theorem 1.5 in math.LO/9910161. Assuming CH any aleph_1$-free splitter of cardinality aleph_1 is free over its nucleus as shown in math.LO/9910161. Generally these modules are very close to being free as explained below. This change follows from math.LO/9910161 and is due to an incomplete proof (noticed thanks to Paul Eklof) in the first section of math.LO/9910161. Assuming the negation of CH, in Shelah [Sh:F417] (work in progress) it will be shown that under Martin's axiom these splitters are free indeed. However there are models of set theory having non-free aleph_1-free splitter of cardinality aleph_1.

Explore related subjects

math.LOmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rüdiger Göbel, Saharon Shelah. 2000-09-06. On our paper `Almost Free Splitter', a correction. https://arxiv.org/abs/math/0009063

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Amenable C-sequences and ineffability in L

We give a relative consistency counterexample to the conjecture of Inamdar and Rinot that every regular uncountable cardinal carrying no amenable C-sequence is ineffable in the constructible universe L. The existence of a $Π^2_1$-indescribable cardinal is equiconsistent with the existence of a $Π^2_1$-indescribable cardinal carrying no amenable C-sequence and not ineffable in L. For every ground-model C-sequence on a weakly compact cardinal $κ$, generalized Hechler forcing adds a club D with $D\capβ\subseteq C_β$ at stationarily many inaccessible indices. We use this capture theorem in an iteration of length $κ^+$ after Johnstone's local lottery preparation.

math.LO↗
arXiv2026

Omitting Types for Infinitary Real-Valued Logic: A Topological Perspective

For countable vocabularies, we extend the omitting types theorem for countable fragments of infinitary logic to Keisler's general real-valued structures by proving that the space of Henkin types of each such fragment is Čech-complete. The structures interpret function symbols arbitrarily and predicate symbols by bounded real-valued functions, while each formula carries a bound of its own. This extends the setting of Eagle's theorem from metric structures with a prescribed master metric to general structures whose vocabulary prescribes the sorts and bounds of the symbols. Following Keisler and Miller's treatment of logic without equality, we build models from closed terms. Morley's Baire category method then gives every satisfiable theory on a countable fragment a model with countable sorts omitting any countable family of types whose closed-term instances have empty interior in the space of Henkin types satisfying the theory.

math.LO↗
arXiv2026

Friedman--Stanley cofinality without Borel completeness

Friedman and Stanley asked whether an invariant Borel class of countable structures must be Borel complete whenever every countable level of their hierarchy reduces to it. We construct a counterexample in one fixed countable language. The structural counterexample and its principal supporting results have been formalized in Lean~4 and it is available on github.

math.LO↗