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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 127 records · Page 7Linked to original sources
arXiv2022

Universality: new criterion for non-existence

We find new "reasons" for a class of models for not having a universal model in a cardinal $λ$. This work, though it has consequences in model theory, is really in combinatorial set theory. We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of $T_{\rm ceq}$ (essentially another representation of $T_{\rm feq}$ which was already considered but the proof with $T_{\rm ceq}$ is more transparent). Models of $T_{\rm ceq}$ consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in $λ$. This work is continued in [Sh:F2071].

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arXiv2022

Beginning of stability theory for Polish Spaces

We consider stability theory for Polish spaces and more generally for definable structures (say, with elements of a set of reals). We clarify by proving some equivalent conditions for $\aleph_0$-stability. We succeed to prove existence of indiscernibles under reasonable conditions; this gives strong evidence that such a theory exists.

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arXiv2022

Many forcing axioms for all regular uncountable cardinals

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.

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arXiv2022

First-Order Aspects of Coxeter Groups

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. $λ$-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and $2$-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if the monoid $Sim(W, S)$ of $S$-self-similarities of $W$ is finitely generated, then $W$ is a prime model of its theory. Thirdly, we prove that in reflection independent RACGs of finite rank the Coxeter elements are type-determined. We then move to $2$-spherical Coxeter groups, proving that if $(W, S)$ is irreducible, $2$-spherical even and not affine, then $W$ is a prime model of its theory, and that if $W_Γ$ and $W_Θ$ are as in the previous sentence, then $W_Γ$ is elementary equivalent to $W_Θ$ if and only if $Γ\cong Θ$, thus solving the elementary equivalence problem for most of the $2$-spherical Coxeter groups. In the last part of the paper we focus on model theoretic applications of the notion of reflection length from Coxeter group theory, proving in particular that affine Coxeter groups are not connected.

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arXiv2022

On the bounding, splitting, and distributivity numbers

The cardinal invariants $ \mathfrak h, \mathfrak b, \mathfrak s$ of $\mathcal P (ω)$ are known to satisfy that $ω_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak h$ and show that it can be less than $\mathfrak s$. The key method is to utilize finite support matrix iterations of ccc posets following \cite{BlassShelah}.

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arXiv2022

Lower bounds on coloring numbers from hardness hypotheses in PCF theory

We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most beth_omega (nu)" proved by Kojman [6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing argument. Similarly, a better upper bound than beth_omega (nu) in this statement implies stronger forms of the RGCH theorem hold, whose consistency and the consistency of their negations are wide open. Thus, the optimality of Kojman's upper bound is a purely cardinal arithmetic problem, and, as discussed below, is hard to decide.

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arXiv2022

Boolean Types in Dependent Theories

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

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arXiv2022

Universal graphs between a strong limit singular and its power

The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $κ$ is supercompact and the cardinals $θ< κ$, $λ> κ$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(κ) = θ$, the Singular Cardinal Hypothesis fails at $κ$ and there exists a universal graph in cardinality $λ\in (κ,2^κ)$.

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arXiv2021

Aristotelian poetry

Jing Zhang proved the consistency of $\binom{ω_2}{ω_1}\rightarrow\binom{n}{ω_1}_ω$ for every $n\inω$ with the negative relation $\binom{ω_2}{ω_1}\nrightarrow\binomω{ω_1}_ω$. We reduce the consistency strength of this statement to an $ω_1$-Erdos cardinal.

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arXiv2021

Hanf numbers for Extendibility and related phenomena

This paper contains portions of Baldwin's talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of $L_{ω_1,ω}$ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.

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arXiv2021

Maximal models up to the first measurable in ZFC

Theorem: There is a {\em complete sentence} $ϕ$ of $L_{ω_1,ω}$ such that $ϕ$ has maximal models in a set of cardinals $λ$ that is cofinal in the first measurable $μ$ while $ϕ$ has no maximal models in any $χ\geq μ$.

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arXiv2021

ZF+DC+AX$_4$

We consider mainly the following version of set theory:"ZF + DC and for every $λ,λ^{\aleph_0}$ is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a sequence $\barδ= \langle δ_s:s \in Y\rangle,\textrm{cf}(δ_s)$ large enough compared to $Y$, we can prove the pcf theorem with minor changes (using true cofinalities not the pseudo one). We then deduce the existence of covering numbers and define and prove existence of truly successor cardinals. From this we show that some diagonalization arguments (more specifically some black boxes and consequence) on Abelian groups. We end by showing some such consequences hold in ZF above.

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arXiv2021

Definable groups for dependent and 2-dependent theories

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\barκ$ type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is $\leq 2^{|T|}$. We then deal with 2-dependent theories, a wider class of first order theories.

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arXiv2021

Universal Structures

We deal with the existence of universal members in a given cardinality for several classes. First we deal with classes of Abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality. Second, we then deal with (variants of) the oak property (from a work of Dzamonja and the author), a property of complete first order theories, sufficient for the non-existence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for quantifier free formulas) and deal more with the existence of universals.

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arXiv2021

Existentially closed locally finite groups

We investigate this class of groups originally called ulf (universal locally finite groups) of cardinality $λ$. We prove that for every locally finite group $G$ there is a canonical existentially closed extention of the same cardinality, unique up to isomorphism and increasing with $G$. Also we get, e.g. existence of complete members (i.e. with no non-inner automorphisms) in many cardinals (provably in ZFC). We also get a parallel to stability theory in the sense of investigating definable types.

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arXiv2021

Isomorphic limit ultrapowers for infinitary logic

The logic L^1_θintroduced in [Sh:797]; it is the maximal logic below L_theta theta in which a well ordering is not definable. We investigate it for theta a compact cardinal. We prove it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are L^1_theta-equivalent iff for some omega-sequence of theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.

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arXiv2021

On failure of 0-1 laws

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 inductive logic.

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