When first order T has limit models
We to a large extent sort out when does a (first order complete theory) T have a superlimit model in a cardinal lambda . Also we deal with relation notions of being limit.
math.LO↗
arXiv subjects
Saharon Shelah
Publications and source records attributed to Saharon Shelah.
PCF and Abelian Groups
We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that "almost always" there are F subseteq lambda^kappa which are quite free and has black boxes. The "almost always" means that there are strong restrictions on cardinal arithmetic if the universe fails this, the restrictions are "everywhere". Also we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer.
math.LO↗
A comment on "p<t"
Dealing with the cardinal invariants p and t of the continuum we prove that m=p=aleph_2 -> t = aleph_1. In other words if MA_{aleph_1} (or a weak version of this) then (of course aleph_2 <= p <= t and) p = aleph_2 -> p = t . This is based on giving a consequence of p<t.
math.LO↗
Dependent T and existence of limit models
Does the class of linear orders have (one of the variants of) the so called (lambda, kappa)-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive, i.e. existence results. More generally, letting T be a complete first order theory and for simplicity assume G.C.H., for regular lambda > kappa > |T| does T have (variants of) a (lambda,kappa)-limit models, except for stable T? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent T we get negative, i.e. non-existence results. We deal more with linear orders.
math.LO↗
Theories with EF-Equivalent Non-Isomorphic Models
Our "long term and large scale" aim is to characterize the first order theories T (at least the countable ones) such that: for every ordinal alpha there lambda,M_1,M_2 such that M_1,M_2 are non-isomorphic models of T of cardinality lambda which are EF_{alpha, lambda}-equivalent. We expect that as in the main gap we get a strong dichotomy, so in the non-structure side we have stronger, better examples, and in the structure side we have a parallel of [Sh:c,XIII]. We presently prove the consistency of the non-structure side for T which is aleph_0-independent (= not strongly dependent), even for PC(T_1, T).
math.LO↗
How you are with $\mathfrak{s}$ and $\mathfrak{r}$?
We prove that if there is a real-valued measurable cardinal then the splitting number is $\aleph_1$. Likewise, if the continuum is real-valued measurable then the reaping number equals the continuum.
math.LO↗
Failure of 0-1 law for sparse random graph in strong logics
Let $α\in(0,1)_\mathbb{R}$ be irrational and $G_n = G_{{n, 1/n}^α}$ be the random graph with edge probability $1/n^α$; we know that it satisfies the 0-1 law for first order logic. We deal with the failure of the 0-1 law for stronger logics: $\mathbb{L}_{ \infty, k}, k$ large enough and the LFP, least fix point logic.
math.LO↗
The Automorphism Group of Hall's Universal Group
We study the automorphism group of Hall's universal locally finite group $H$. We show that in $Aut(H)$ every subgroup of index $< 2^ω$ lies between the pointwise and the setwise stabilizer of a unique finite subgroup $A$ of $H$, and use this to prove that $Aut(H)$ is complete. We further show that $Inn(H)$ is the largest locally finite normal subgroup of $Aut(H)$. Finally, we observe that from the work of [Sh:312] it follows that for every countable locally finite $G$ there exists $G \cong G' \leq H$ such that every $f \in Aut(G')$ extends to an $\hat{f} \in Aut(H)$ in such a way that $f \mapsto \hat{f}$ embeds $Aut(G')$ into $Aut(H)$. In particular, we solve the three open questions of Hickin on $Aut(H)$ from [3], and give a partial answer to Question VI.5 of Kegel and Wehrfritz from [6].
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Madness and regularity properties
Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $ω^ω$-bounding sufficiently absolute forcing notions $\mathbb Q$.
math.LO↗
The last forcing standing with diamonds
This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+.
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No Uncountable Polish Group Can be a Right-Angled Artin Group
We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0 < k < ω$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < ω$ and $x^k = y$, then $x = e$. In particular, the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes results from [3] and [5], where this is proved for free and free Abelian uncountable groups.
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A strong failure of aleph_0-stability for atomic classes
We study classes of atomic models At_T of a countable, complete first-order theory T . We prove that if At_T is not pcl-small, i.e., there is an atomic model N that realizes uncountably many types over pcl(a) for some finite tuple a from N, then there are 2^aleph1 non-isomorphic atomic models of T, each of size aleph1.
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A parallel to the null ideal for inaccessible lambda. Part I
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an positive = existence answer for lambda a "mild" large cardinals: the weakly compact ones. We apply this to get consistency results on cardinal invariants for such lambda's. We shall deal with other cardinals more properties related forcing notions in a continuation.
math.LO↗
Mad families and non-meager filters
We prove the consistency of ZF+DC+"there are no mad families"+"there exists a non-meager filter on $ω$" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.
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On cuts in ultraproducts of linear orders I
For an ultrafilter $D$ on a cardinal $κ,$ we wonder for which pair $(θ_1, θ_2)$ of regular cardinals, we have: for any $(θ_1+θ_2)^+-$saturated dense linear order $J, J^κ/ D$ has a cut of cofinality $(θ_1, θ_2).$ We deal mainly with the case $θ_1, θ_2 > 2^κ.$
math.LO↗
Maximal independent sets in Borel graphs and large cardinals
We construct a Borel graph G such that ZF+DC+"There are no maximal independent sets in G" is equiconsistent with ZFC+"There exists an inaccessible cardinal".
math.LO↗
Small-large subgroups of the reals
We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgrooup of R, while it isconsistent there there is no non-null meager subgroup of R.
math.LO↗