arXiv subjects
Publications and source records attributed to Saharon Shelah.
We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-$ω_2$ countable support iteration of restricted Mathias forcing over a ground model satisfying $\textsf{CH}$, there is a unique $Q$-point up to isomorphism. In particular, it is consistent that there is only one $Q$-point while there are $2^{\mathfrak{c}}$-many near-coherence classes of ultrafilters.
Let $ κ, θ< λ$ be cardinals, with $λ$ and $κ$ regular. Concentrating on a simple case, we say that the triple $(λ,κ,θ)$ has a Super Black Box when the following holds. For some stationary $S \subseteq \{δ< λ: cf(δ) = κ\}$ and $\overline C = \langle C_δ: δ\in S \rangle$, where $C_δ$ is a club of $δ$ of order type $κ$, for every coloring $\overline F = \langle F_δ: δ\in S \rangle$ with $F_δ: {}^{C_δ}λ\to θ$, there exists $\langle c_δ: δ\in S\rangle \in {}^S\!θ$ such that for every $f : λ\to θ$, for stationarily many $δ\in S$, we have $F_δ(f \upharpoonright C_δ) = c_δ$. In an earlier work, it was proved (along with much more) that for a class of cardinals $λ$ this holds for many pairs $(κ,θ)$. E.g.~$κ< \aleph_ω$ is large enough, and $\beth_ω(θ) < λ$. However, the most interesting cases (at least with regards to Abelian groups) are $κ= \aleph_0,\aleph_1$ (which have not been covered yet). Here we restrict ourselves to the case where $\overline F$ is a {so-called} \emph{continuous coloring}, which includes the case where $F_δ$ is computed from some $$ \big\langle F_{δ,β}'(f \upharpoonright (C_δ\cap β)) : β\in C_δ\big\rangle. $$ This covers the cases we have in mind. We mainly prove results without any other caveats: e.g. For every regular $κ$ and $θ$ there exists such a $λ$. We also deal with having multiple {$\bar C$-s}, and the existence of quite free subsets of ${}^κμ$.
Assuming Gödel's axiom of constructibility $V=L$, we construct a $χ$-free abelian group $G$ of singular cardinality for some suitable cardinal $χ$ which is regular and uncountable, equipped with the property that for every nontrivial subgroup $G' \subseteq G$ of smaller cardinality, $Hom(G',\mathbb{Z}) \neq 0$, while $Hom(G,\mathbb{Z}) = 0$. This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor $Hom(-,\mathbb{Z})$.
In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete, in fact, more strongly, we can $\mathfrak{L}_{ω_1, ω}$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\mathrm{Th}(\mathbb{Z}^{(ω)})$ is a complete analytic quasi-order.
Characteristic earlier results were of the form CON$(2^{\aleph_0} \to [λ]^2_{n, 2})$, with $2^{\aleph_0} $ an ex-large cardinal, in the best case the first weakly Mahlo cardinal. Characteristic new results are CON$((2^{\aleph_0} = \aleph_m) + \aleph_l \to [\aleph_k]^2_{n, 2})$, for suitable $k < l < m$. So we improve in three respects: the continuum may be small (e.g. not a weakly Mahlo), we use no large cardinal, and the cardinals $λ$ involved are $ < 2^{\aleph_0}$ after the forcing.
We look for partition theorems for large subtrees for suitable uncountable trees and colourings. We concentrate on sub-trees of $^{κ\ge} 2$ expanded by a well ordering of each level. Unlike earlier works, we do not ask the embedding to preserve the height of the tree but the equality of levels is preserved. We get consistency results without large cardinals. The intention is to apply it to model theoretic problems.
We contribute to the study of $\aleph_1$-dense sets of reals, a mainstay in set theoretic research since Baumgartner's seminal work in the 70s. In particular, we show that it is consistent with $\textsf{MA}$ that there exists an $\aleph_1$-dense set of reals $A$ so that, in any cardinal-preserving generic extension by a forcing of size $\aleph_1$, $A$ and $A^*$ do not contain uncountable subsets which are order isomorphic. This strengthens a result of Avraham and the second author and yields a different proof of a theorem of Moore and Todorcevic.
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that there is a rich array of associative sums different from the usual sum $+$ and its dual. The simplest of these sums arise from what we call sum-generating classes of linear orders. These classes determine canonical decompositions of every linear order into left and right halves. We study the structural and algebraic properties of these classes along with the sums they generate. We then turn our attention to commutative sums on various subclasses of the linear orders. For this, we introduce the notion of a complicated class of linear orders and show that over such classes sums can be constructed in a very flexible way. Using this construction, we prove the existence of associative sums lacking the structural properties of the usual sum. Along the way, we characterize the associative and commutative sums on the ordinals.
We prove that every abstract elementary class (a.e.c.) with LST number $κ$ and vocabulary $τ$ of cardinality $\leq κ$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(κ)^{+++},κ^+}(τ)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_λ$.}
Abstractly, the generic extensions after $\aleph_ω$-many Cohen reals and $\aleph_{ω+1}$-many Cohen reals must be different for reasons of uniform density the relevant Boolean algebras. Nevertheless this is not satisfying and it would be nice to pin the difference between the two models down to some mathematical or combinatorial principle. In this paper we provide such a principle.
We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work broadens the scope to include groups with torsion. Our main focus is on p-groups, for which we formulate and prove the Trivial Duality Conjecture. Key tools in our analysis include the multi black box method and the application of specific homological properties of relative trees.
For a given stationary set $S$ of countable ordinals we prove (in $\mathbf{ZFC}$) that the assertion "every $S$-ladder system has $\aleph_0$-uniformization" is equivalent to "every strongly $\aleph_1$-free abelian group of cardinality $\aleph_1$ with non-freeness invariant $\subseteq S$ is $\aleph_1$-coseparable, i.e. Ext$(G, \oplus_{i=0}^{\infty} \mathbb Z)=0$ (in particular Whitehead, i.e.\ Ext$(G, \mathbb Z)=0$)". This solves problems B3 and B4 from Eklof and Mekler's monograph.
In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using the reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.
We consider the following property of a first order theory T with a distinguished unary predicate P: every model of the theory of P occurs as the P-part of some model of T. We call this property the Gaifman property. Gaifman conjectured that if T is relatively categorical over P, then it has the Gaifman property. We propose a generalized version of this conjecture: if T fails the Gaifman property, then it exhibits non-structure over P, i.e., has many non-isomorphic models over P in many cardinalities. We address this conjecture for countable theories. Motivated by ideas from Classification Theory, we separate this conjecture into two parts: 1) stability over P (a structure property of theories) implies the Gaifman property, and 2) instability over P implies non-structure. In this paper prove the first part of this conjecture. In fact, we prove a stronger statement: an appropriate version of stability implies higher stable amalgamation properties.
We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups $G$ of size $< \mathfrak{p}$ with infinite $\mathrm{Tor}_p(G)$, and that in particular there are no infinite reduced abelian $p$-groups of size $< \mathfrak{p}$. Secondly, we prove that if $2^{\aleph_0} < λ< λ^{\aleph_0}$, and $G$ is abelian of size $λ$, then $G$ is not co-Hopfian. Finally, we prove that for every cardinal $λ$ there is a torsion-free abelian group $G$ of size $λ$ which is absolutely Hopfian, i.e., $G$ is Hopfian and $G$ remains Hopfian in every forcing extensions of the universe.
For which (first-order complete, usually countable) $T$ do there exist non-isomorphic models of $T$ which become isomorphic after forcing with a forcing notion $\mathbb{P}$? Necessarily, $\mathbb{P}$ is non-trivial; i.e.~it adds some new set of ordinals. It is best if we also demand that it collapses no cardinal. It is better to demand on the one hand that the models are non-isomorphic, and even \emph{far} from each other (in a suitable sense), but on the other hand, $\mathcal{L}$-equivalent in some suitable logic $\mathcal{L}$. We give sufficient conditions: for theories with the independence property, we proved this when $\mathbb{P}$ adds no new $ω$-sequence. We may prove it ``for some $\mathbb{P}$," but better would be for some specific forcing notions. Best would be to characterize the pairs $(T,\mathbb{P})$ for which we have such models. The results say (e.g.) that there are models $M_1,M_2$ which are not isomorphic (and even \emph{far} from being isomorphic, in a rigorous sense) which become isomorphic when we extend the universe by adding a new branch to the tree $({}^{θ>}2,\lhd)$. We shall mention some specific choices of $\mathbb{P}$: mainly $({}^{θ>}2,\lhd)$ with $θ= θ^{<θ}$. That is, the reader just has to agree that starting with a universe $\mathbf{V}$ of set theory (i.e.~a model of $ZFC$) and a quasiorder $\mathbb{P}$, there are new directed $\mathbf{G} \subseteq \mathbb{P}$ meeting every dense subset of $D$ of $\mathbb{P}$, a universe $\mathbf{V}[\mathbf{G}]$ (so {it} satisfies $ZFC$) of which the original $\mathbf{V}$ is a transitive subclass. We may say that $\mathbf{V}[\mathbf{G}]$ (also denoted $\mathbf{V}^\mathbb{P}$) is the universe obtained by forcing with $\mathbb{P}$. This work does not require any serious knowledge of forcings, nor of stability theory. This is part of the classification and {Main Gap} program.
Using creature technology, we construct families of Suslin ccc non-sweet forcing notions $\mathbb Q$ such that $ZFC$ is equiconsistent with $ZF+$"every set of reals equals a Borel set modulo the $(\leq \aleph_1)$-closure of the null ideal associated with $\mathbb Q$"+"there is an $ω_1$-sequence of distinct reals". This answers a question of the second author and Kellner. As an application of independent interest, we also show how our forcing adds a new $Π^1_2$ singleton over $L$ without relying on $L$-combinatorics.
We prove that any ordered field can be extended to one for which every decreasing sequence of bounded closed intervals, of any length, has a nonempty intersection; equivalently, there are no Dedekind cuts with equal cofinality from both sides. Here we strengthen the results from the published version.