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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 73 records · Page 4Linked to original sources
arXiv2023

Anti-Classification Results for Rigidity Conditions in Abelian and Nilpotent Groups

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian groups are complete co-analytic in the space of countable torsion-free abelian groups ($\mathrm{TFAB}_ω$); (2) the Hopfian groups are complete co-analytic in $\mathrm{TFAB}_ω$; (3) the co-Hopfian groups are complete co-analytic in the space of countable $2$-nilpotent groups. In combination with our result from [13, S5], which shows that the endo-rigid abelian groups are complete co-analytic in $\mathrm{TFAB}_ω$, this shows that four major notions of rigidity from (abelian) group theory are as complex as possible as co-analytic problems. Further, the second and third theorem above solve two open questions of Thomas from [18], who asked this for the space of all countable groups. We leave open the question of whether the co-Hopfian mixed abelian groups are complete co-analytic in the space of countable abelian groups, but we reduce the problem to a concrete question on profinite groups, showing that if $G$ is a countable co-Hopfian abelian reduced group, then, for every prime number $p$, the torsion subgroup $\mathrm{Tor}_p(G)$ of $G$ is finite and $G$ embeds in the profinite group $ \prod_{p \in \mathbb{P}} \mathrm{Tor}_p(G)$.

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arXiv2023

More Ramsey theory for highly connected monochromatic subgraphs

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting that if we color edges of the complete graph we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák and Shelah are answered by showing that assuming the consistency of suitable large cardinals the following are relatively consistent with $\mathsf{ZFC}$: $κ\to_{hc} (κ)^2_ω$ for every regular cardinal $κ\geq \aleph_2$ and $\neg\mathsf{CH}+ \aleph_2 \to_{hc} (\aleph_1)^2_ω$. Building on a work of Lambie-Hanson, we also show that $\aleph_2 \to_{hc} [\aleph_2]^2_{ω,2}$ is consistent with $\neg\mathsf{CH}$. To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

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arXiv2023

Co-Hopfian and boundedly endo-rigid mixed abelian groups

For a given cardinal $λ$ and a torsion abelian group $K$ of cardinality less than $λ$, we present, under some mild conditions (for example $λ=λ^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of cardinality $λ$ with $Tor(G)=K$. Essentially, we give a complete characterization of such pairs $(K, λ)$. Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals $λ> 2^{\aleph_{0}}$ for which there is a co-Hopfian abelian group of size $λ$.

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arXiv2023

Adding highly undefinable sets over $L$

We sow that there exists a generic extension of the Gödel's constructible universe in which diamond holds and there exists a subset $Y \subseteq ω_1$ such that for stationary many $δ< ω_1,$ the set $Y \cap δ$ is not definable in the structure $(L_{F(δ)}, \in)$, where $F(δ) > δ$ is the least ordinal such that $L_{F(δ)}\models$``$δ$ is countable''.

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arXiv2023

The Slicing Axioms

We introduce the family of axioms, denoted $\operatorname{Slice}_κ$, that claim the existence of strictly increasing decompositions of the form $$2^δ=\bigcup_{α<κ} 2^δ\cap M_α,$$ where $δ<κ$, and $\{M_α|\; α<κ\}$ is a $\subseteq$-increasing sequence of transitive models of set theory. We study compatibility of these axioms with versions of Martin's Axiom, and in particular show that $\operatorname{Slice}$ is compatible only with some very weak form of $MA$.

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arXiv2023

When does $\aleph_1$-categoricity imply $ω$-stability?

For an $\aleph_1$-categorical atomic class, we clarify the space of types over the unique model of size $\aleph_1$. Using these results, we prove that if such a class has a model of size $\beth_1^+$ then it is $ω$-stable.

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arXiv2023

Many partition relations below density

We force $2^λ$ to be large and for many pairs in the interval $(λ,2^λ)$ a stronger version of the polarized partition relations hold. We apply this toproblem in general topology

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arXiv2023

Atomic saturation of reduced powers

Our aim was to generalize some theorems about the saturation of ultra-powers to reduced powers. Naturally, we deal via saturation for types consisting of atomic formulas. We succeed to generalize the theory of dense linear is maximal and so is any pair (T, Delta) which is SOP_3 (Delta consists of atomic or conjunction of atomic formulas). However, SOP_2 is not enough, so the p equals t theorem cannot be generalized in this case. Similarly the unique dual of cofinality.

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arXiv2023

Generalizing random real forcing for inaccessible cardinals

The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for $\aleph^{\aleph_0}_0$; in spite of this similarity, the Cohen forcing and Random Real forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for $λ^λ$ while $λ> \aleph_0$, corresponding to an extension for the meagre sets, while the Random real forcing didn't see to have a natural generalization, as Lebesgue measure doesn't have a generalization for space $λ^λ$ while $λ> \aleph_0$. Shelah found a forcing resembling the properties of Random Real Forcing for $λ^λ$ while $λ$ is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for $λ^λ$ while $λ$ is an inaccessible cardinal; this forcing preserves cardinals and cofinalities, however unlike Cohen forcing, does not add undominated reals.

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arXiv2023

Model Theory for a Compact Cardinal

We like to develop model theory for $T$, a complete theory in $\mathbb{L}_{θ,θ}(τ)$ when $θ$ is a compact cardinal. By [Sh:300a] we have bare bones stability and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) we restrict ourselves to ``$D$ a $θ$-complete ultrafilter on $I$, probably $(I,θ)$-regular". The basic theorems work, but can we generalize deeper parts of model theory? In particular can we generalize stability enough to generalize [Sh:c, Ch.VI]? We prove that at least we can characterize the $T$'s which are minimal under Keisler's order, i.e. such that $\{D:D$ is a regular ultrafilter on $λ$ and $M \models T \Rightarrow M^λ/D$ is $λ$-saturated$\}$. Further we succeed to connect our investigation with the logic $\mathbb{L}^1_{< θ}$ introduced in [Sh:797]: two models are $\mathbb{L}^1_{< θ}$-equivalent iff \, for some $ω$- sequence of$θ$-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.

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arXiv2023

Universally measurable sets may all be Delta^1_2

We produce a forcing extension of the constructible universe $\bL$ in which every universally measurable set of reals is $\uTDelta^{1}_{2}$, partially answering question CG from David Fremlin's problem list. The analogous result for category holds in the same model.

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arXiv2023

Halfway New Cardinal Characteristics

Based on the well-known cardinal characteristics $\mathfrak{s}$, $\mathfrak{r}$ and $\mathfrak{i}$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties of infinite sets. We prove several bounds and consistency results, e. g. the consistency of $\mathfrak{s} < \mathfrak{s}_{1/2}$, as well as several results about possible values of $\mathfrak{i}_{1/2}$.

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arXiv2023

Tiltan and superclub

We force superclub with an arbitrary large value of cov($\mathscr{M}$). We force tiltan with an arbitrary large value of add($\mathscr{M}$). Finally, we obtain a negative square bracket relation from superclub.

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arXiv2023

Implications of Ramsey Choice Principles in ZF

The Ramsey Choice principle for families of $n$-element sets, denoted $\mathrm{RC}_n$, states that every infinite set $X$ has an infinite subset $Y\subseteq X$ with a choice function on $[Y]^n := \{z\subseteq Y : |z| = n\}$. We investigate for which positive integers $m$ and $n$ the implication $\mathrm{RC}_m \Rightarrow \mathrm{RC}_n$ is provable in ZF. It will turn out that beside the trivial implications $\mathrm{RC}_m \Rightarrow \mathrm{RC}_m$, under the assumption that every odd integer $n>5$ is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in ZF is $\mathrm{RC}_2 \Rightarrow \mathrm{RC}_4$.

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arXiv2023

General non-structure theory and constructing from linear orders

The theme of the first two sections, is to prepare the framework of how from a ``complicated'' family of so called index models $I \in K_1$ we build many and/or complicated structures in a class $K_2$. The index models are characteristically linear orders, trees with $κ+1$ levels (possibly with linear order on the set of successors of a member) and linearly ordered graphs; for this we formulate relevant complicatedness properties (called bigness). In the third section we show stronger results concerning linear orders. If for each linear order $I$ of cardinality $λ> \aleph_0$ we can attach a model $M_I \in K_λ$ in which the linear order can be embedded such that for enough cuts of $I$, their being omitted is reflected in $M_I$, then there are $2^λ$ non-isomorphic cases. We also do the work for some applications.

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arXiv2023

AEC: weight and $p$-simplicity

Part I: We would like to generalize imaginary elements, weight of ${\rm ortp}(a,M,N),{\mathbf P}$-weight, ${\mathbf P}$-simple types, etc. from [Sh:c, Ch.III,V,\S4] to the context of good frames. This requires allowing the vocabulary to have predicates and function symbols of infinite arity, but it seemed that we do not suffer any real loss. Part II: become [1238] Good frames were suggested in [Sh:h] as the (bare bones) right parallel among a.e.c. to superstable (among elementary classes). Here we consider $(μ,λ,κ)$-frames as candidates for being the right parallel to the class of $|T|^+$-saturated models of a stable theory (among elementary classes). A loss as compared to the superstable case is that going up by induction on cardinals is problematic (for cardinals of small cofinality). But this arises only when we try to lift. But this context we investigate the dimension. Part III: become [1239] In the context of Part II, we consider the main gap problem for the parallel of somewhat saturated model; showing we are not worse than in the first order case.

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