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Martin Goldstern

Publications and source records attributed to Martin Goldstern.

At least 19 recordsLinked to original sources
arXiv2024

Preservation of splitting families and cardinal characteristics of the continuum

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń's diagram, $\mathfrak{m}(2\text{-Knaster})$, $\mathfrak{p}$, $\mathfrak{h}$, the splitting number $\mathfrak{s}$ and the reaping number $\mathfrak{r}$.

math.LO↗
arXiv2021

Controlling cardinal characteristics without adding reals

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}κ$-sequences (for some regular $κ$). As an application, we show that consistently the following cardinal characteristics can be different: The ("independent") characteristics in Cichoń's diagram, plus $\aleph_1<\mathfrak m<\mathfrak p<\mathfrak h<\mathrm{add}(\mathcal{N})$. (So we get thirteen different values, including $\aleph_1$ and continuum). We also give constructions to alternatively separate other MA-numbers (instead of $\mathfrak m$), namely: MA for $k$-Knaster from MA for $k+1$-Knaster; and MA for the union of all $k$-Knaster forcings from MA for precaliber.

math.LO↗
arXiv2020

Controlling classical cardinal characteristics while collapsing cardinals

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $λ>κ$ to $κ$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cichoń's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cichoń's Maximum construction, which avoids large cardinals.)

math.LO↗
arXiv2020

Cichoń's maximum without large cardinals

Cichoń's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $σ$-compact subsets of the irrationals. It is consistent that all entries of Cichoń's diagram are pairwise different (apart from $\textrm{add}(\mathcal{M})$ and $\textrm{cof}(\mathcal{M})$, which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions.

math.LO↗
arXiv2019

Stranger Things about Forcing without AC

Typically, set theorists reason about forcing constructions in the context of ZFC. We show that without AC, several simple properties of forcing posets fail to hold, one of which answers Miller's question from arXiv:0704.3998.

math.LO↗
arXiv2019

Cichoń's maximum

Assuming four strongly compact cardinals, it is consistent that all entries in Cichoń's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]

math.LO↗
arXiv2019

Set-Theoretic Blockchains

Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.

math.LO↗
arXiv2018

The Higher Cichoń Diagram

For a strongly inacessible cardinal $κ$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}κ$-box product topology - the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799) - the ideal of nowhere stationary subsets of a (naturally defined) stationary set $S_{\rm pr}^κ\subseteq κ$. In particular, we analyse the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case ($κ=ω$) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null)$\le$non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with $\mathord<κ$-support (and a strong "Knaster" version of $κ^+$-cc) and one iteration with ${\le}κ$-support (and a version of $κ$-properness).

math.LO↗
arXiv2015

New reals: Can live with them, can live without them

We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation," "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.

math.LO↗
arXiv2015

Creature forcing and five cardinal characteristics in Cichoń's diagram

We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]

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↗
arXiv2011

A closed algebra with a non-Borel clone and an ideal with a Borel clone

Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest that it is complete coanalytic, but which turns out to be a rather simple Borel set.

math.RA↗
arXiv2011

Clones above the unary clone

Let c be the cardinality of the continuum. We give a family of pairwise incomparable clones (on a countable base set) 2^c members, all with the same unary fragment, namely the set of all unary operations. We also give, for each n, a family of 2^c clones all with the same n-ary fragment, and all containing the set of all unary operations.

math.RA↗
arXiv2009

Optimal and better transport plans

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value infty. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=infty} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a "better" notion of optimality called robust optimality.

math.OC↗
arXiv2008

Clones from ideals

On an infinite base set X, every ideal of subsets of X can be associated with the clone of those operations on X which map small sets to small sets. We continue earlier investigations on the position of such clones in the clone lattice.

math.RA↗
arXiv2008

Ideal clones: Solution to a problem of Czedli and Heindorf

Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In a 2001 article, G. Czedli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question.

math.RA↗