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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 271 records · Page 15Linked to original sources
arXiv2015

Two inequalities between cardinal invariants

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $ω$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove in sharp contrast to the case when $κ= ω$, that if $κ$ is any regular uncountable cardinal, then ${\mathfrak{s}}_κ \leq {\mathfrak{b}}_κ$.

math.LO↗
arXiv2015

The left side of Cichoń's diagram

Using a finite support iteration of ccc forcings, we construct a model of $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})=\mathfrak{c}$.

math.LO↗
arXiv2015

Forcing a countable structure to belong to the ground model

Suppose that $P$ is a forcing notion, $L$ is a language (in $V$), $\dotτ$ a $P$-name such that $P\Vdash$ "$\dotτ$ is a countable $L$-structure". In the product $P\times P$, there are names $\dot{τ_{1}},\dot{τ_{2}}$ such that for any generic filter $G=G_{1}\times G_{2}$ over $P\times P$, $\dotτ_{1}[G]=\dotτ[G_{1}]$ and $\dotτ_{2}[G]=\dotτ[G_{2}]$. Zapletal asked whether or not $P \times P \Vdash \dotτ_{1}\cong\dotτ_{2}$ implies that there is some $M\in V$ such that $P \Vdash \dotτ\cong\check{M}$. We answer this negatively and discuss related issues.

math.LO↗
arXiv2015

Constructing many atomic models in $\aleph_1$

We introduce the notion of pseudo-algebraicity to study atomic models of first order theories (equivalently models of a complete sentence of $L_{ω_1,ω}$. Theorem: Let $T$ be any complete first-order theory in a countable language with an atomic model. If the pseudo-minimal types are not dense, then there are $2^{\aleph_1}$ pairwise non-isomorphic atomic models of $T$, each of size $\aleph_1$.

math.LO↗
arXiv2015

On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $σ$-linked partial orderings. This answers an old question of the first author about the relative strength of Martin's axiom for $σ$-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprans and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph_1$ property of a partial ordering while preserving its ccc-ness.

math.LO↗
arXiv2014

On the number of Dedekind cuts and two-cardinal models of dependent theories

For an infinite cardinal $κ$, let $dedκ$ denote the supremum of the number of Dedekind cuts in linear orders of size $κ$. It is known that $κ<dedκ\leq 2^κ$ for all $κ$ and that $dedκ<2^κ$ is consistent for any $κ$ of uncountable cofinality. We prove however that $2^κ\leq ded ( ded ( ded ( dedκ)))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.

math.LO↗
arXiv2014

On embedding certain partial orders into the P-points under RK and Tukey reducibility

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin's axiom for $σ$-centered posets. In his 1973 paper he showed under this assumption that both $ω_{1}$ and the reals can be embedded. This result was later repeated for the coarser notion of Tukey reducibility. We prove in this paper that Martin's axiom for $σ$-centered posets implies that every partial order of size at most continuum can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility.

math.LO↗
arXiv2014

Monotone hulls for N cap M

Using the method of decisive creatures (math.LO/0601083) we show the consistency of "there is no increasing omega_2 --chain of Borel sets and non(N)=non(M)= omega_2=2^omega". Hence, consistently, there are no monotone hulls for the ideal M cap N . This answers Balcerzak and Filipczak. Next we use FS iteration with partial memory to show that there may be monotone Borel hulls for the ideals M, N even if they are not generated by towers.

math.LO↗
arXiv2014

Rigidity of continuous quotients

We study countable saturation of the metric reduced products and introduce continuous fields of metric models indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand--Naimark duality we conclude that the assertion that the \vCech--Stone remainder of the half-line has only trivial automorphisms is independent from ZFC. The consistency of this statement follows from Proper Forcing Axiom and this is the first known example of a connected space with this property.

math.LO↗
arXiv2014

P-NDOP and P-decompositions of aleph_epsilon-saturated models of superstable theories

Assume a complete superstable theory is superstable, and let P be a class of regular types, typically closed under automorphisms of the monster and non-orthogonality. We define the notion of P-NDOP and prove the existence of P-decompositions and derive an analog of Sh401 for superstable theories with P-NDOP. In this context, we also find a sufficient condition on P-decompositions that imply non-isomorphic models. For this, we investigate natural structures on the types in P\intersect S(M) modulo non-orthogonality.

math.LO↗
arXiv2014

Borel completeness of some aleph_0 stable theories

We study aleph_0-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of lambda-Borel completeness and prove that such theories are lambda-Borel complete. Using this, we conclude that an aleph_0-stable theory has 2^lambda pairwise non-L(infinity,aleph_0) equivalent models of size lambda for all infinite cardinals lambda if and only if T either has eni-DOP or is eni-deep.

math.LO↗
arXiv2014

Universal graphs with a forbidden subgraph: Block path solidity

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.

math.CO↗
arXiv2014

On Borel hull operations

We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets in (in R^n). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the sigma-algebra of sets with the property of Baire.

math.LO↗
arXiv2014

The linear refinement number and selection theory

The \emph{linear refinement number} $\mathfrak{lr}$ is the minimal cardinality of a centered family in $[ω]^ω$ such that no linearly ordered set in $([ω]^ω,\subseteq^*)$ refines this family. The \emph{linear excluded middle number} $\mathfrak{lx}$ is a variation of $\mathfrak{lr}$. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classic combinatorial cardinal characteristics of the continuum. We prove that $\mathfrak{lr}=\mathfrak{lx}=\mathfrak{fd}$ in all models where the continuum is at most $\aleph_2$, and that the cofinality of $\mathfrak{lr}$ is uncountable. Using the method of forcing, we show that $\mathfrak{lr}$ and $\mathfrak{lx}$ are not provably equal to $\mathfrak{d}$, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

math.GN↗
arXiv2014

Mad Spectra

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on omega. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We solve this problem to some extent. What remains open is the possible values of min(A) and max(A).

math.LO↗