Nowhere trivial automorphisms of $P(λ)/[λ]^{<λ}$, for $λ$ inaccessible
If $λ$ is (strongly) inaccessible and $2^λ= λ^+$, then there is a nowhere trivial automorphism of the Boolean algebra $\mathcal P(λ)/[λ]^{<λ}$.
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arXiv subjects
Jakob Kellner
Publications and source records attributed to Jakob Kellner.
On automorphisms of $\mathcal P(λ)/[λ]^{<λ}$
We investigate the statement ``all automorphisms of $\mathcal P(λ)/[λ]^{<λ}$ are trivial''. We show that MA implies the statement for regular uncountable $λ<2^{\aleph_0}$; that the statement is false for measurable $λ$ if $2^λ=λ^+$; and that for ``densely trivial'' it can be forced (together with $2^λ=λ^{++}$) for inaccessible $λ$.
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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 $\mathfrak{r}$.
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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.
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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.)
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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.
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Cichoń's maximum
Assuming four strongly compact cardinals, it is consistent that all entries in Cichoń's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
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Another ordering of the ten cardinal characteristics in Cichoń's diagram
It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) <\mathrm{add}(\mathrm{Meager})=\mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) < \mathrm{non}(\mathrm{Null}) < \mathrm{cof}(\mathrm{Meager})= \mathfrak{d} < \mathrm{cof}(\mathrm{Null}) < 2^{\aleph_0}. \]
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Compact Cardinals and Eight Values in Cichoń's Diagram
Assuming three strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\] Under the same assumption, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
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Pitowsky's Kolmogorovian models and Super-Determinism
In an attempt to demonstrate that local hidden variables are mathematically possible, Pitowsky constructed "spin-$\frac12$ functions" and later "Kolmogorovian models", which employs a nonstandard notion of probability. We describe Pitowsky's analysis and argue (with the benefit of hindsight) that his notion of hidden variables is in fact just super-determinism (and accordingly physically not relevant). Pitowsky's first construction uses the Continuum Hypothesis. Farah and Magidor took this as an indication that at some stage physics might give arguments for or against adopting specific new axioms of set theory. We would rather argue that it supports the opposing view, i.e., the widespread intuition "if you need a non-measurable function, it is physically irrelevant".
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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.
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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 \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
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Borel Conjecture and Dual Borel Conjecture
We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.
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An overview of the proof in "Borel Conjecture and Dual Borel Conjecture"
This note gives an informal overview of the proof in our paper "Borel Conjecture and Dual Borel Conjecture", see arXiv:1105.0823.
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Non-elementary proper forcing
We introduce a simplified framework for ord-transitive models and Shelah's non elementary proper (nep) theory. We also introduce a new construction for the countable support nep iteration.
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Saccharinity
We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal.
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More on the pressing down game
We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: Consistently, there is a nowhere precipitous normal ideal $I$ on $\aleph_2$ such that player nonempty wins the pressing down game of length $\aleph_1$ on $I$ even if player empty starts.
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Creature forcing and large continuum: The joy of halving
For $f,g\inω^ω$ let $c^\forall_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e., for every branch $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. Let $c^\exists_{f,g}$ be the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. We show that it is consistent that $c^\exists_{f_ε,g_ε}=c^\forall_{f_ε,g_ε}=κ_ε$ for continuum many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we introduce a new mixed-limit creature forcing construction.
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