arXiv subjects

Philipp Lücke

Publications and source records attributed to Philipp Lücke.

28 records · Page 2Linked to original sources

arXiv2020

Small models, large cardinals, and induced ideals

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.

math.LO↗

arXiv2018

Characterizing large cardinals through Neeman's pure side condition forcing

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $ω_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings.

math.LO↗

arXiv2018

Partition properties for simply definable colourings

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $ω_1$ for every colouring of the finite sets of countable ordinals that is definable by a $Σ_1$-formula that only uses the cardinal $ω_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given large cardinal and these implications yield natural examples of inaccessible cardinals that possess strong partition properties for $Σ_1$-definable colourings and are not weakly compact. In contrast, we show that $Σ_1$-definability behaves fundamentally different at $ω_2$ by showing that various large cardinal assumptions and \emph{Martin's Maximum} are compatible with the existence of a colouring of pairs of elements of $ω_2$ that is definable by a $Σ_1$-formula with parameter $ω_2$ and has no uncountable homogeneous set. Our results will also allow us to derive tight bounds for the consistency strengths of various partition properties for definable colourings. Finally, we use the developed theory to study the question whether certain homeomorphisms that witness failures of weak compactness at small cardinals can be simply definable.

math.LO↗

arXiv2017

Class forcing, the forcing theorem and Boolean completions

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing. In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

math.LO↗

arXiv2017

$Σ_1(κ)$-definable subsets of $\mathrm{H}(κ^+)$

We study $Σ_1(ω_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $Σ_1$-formula with parameter $ω_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $Σ_1(ω_1)$-definable, the set of all stationary subsets of $ω_1$ is not $Σ_1(ω_1)$-definable and the complement of every $Σ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$ is not $Σ_1(ω_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a $Σ_1(ω_1)$-definable well-ordering of $\mathrm{H}({ω_2})$ and the existence of a $Δ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no $Σ_1(ω_1)$-definable uniformization of the club filter on $ω_1$. Moreover, we prove a perfect set theorem for $Σ_1(ω_1)$-definable subsets of ${}^{ω_1}ω_1$, assuming that there is a measurable cardinal and the non-stationary ideal on $ω_1$ is saturated. The proofs of these results use iterated generic ultrapowers and Woodin's $\mathbb{P}_{\mathrm{max}}$-forcing. Finally, we also prove variants of some of these results for $Σ_1(κ)$-definable subsets of ${}^κκ$, in the case where $κ$ itself has certain large cardinal properties.

math.LO↗

arXiv2017

Squares, ascent paths, and chain conditions

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević's principle $\square(κ)$ implies an indexed version of $\square(κ,λ)$, we show that for all infinite, regular cardinals $λ<κ$, the principle $\square(κ)$ implies the existence of a $κ$-Aronszajn tree containing a $λ$-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which $\aleph_2$-Aronszajn trees exist and all such trees contain $\aleph_0$-ascent paths. Finally, we use our techniques to show that the assumption that the $κ$-Knaster property is countably productive and the assumption that every $κ$-Knaster partial order is $κ$-stationarily layered both imply the failure of $\square(κ)$.

math.LO↗

arXiv2017

Small Embedding Characterizations for Large Cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. In particular, we provide such embedding characterizations also for several large cardinal notions for which no embedding characterizations have been known so far, namely for subtle, for ineffable, and for $λ$-ineffable cardinals. As an application, which we will study in detail in a subsequent paper, we present the basic idea of our concept of internal large cardinals. We provide the definition of certain kinds of internally subtle, internally $λ$-ineffable and internally supercompact cardinals, and show that these correspond to generalized tree properties, that were investigated by Weiß in his [16] and [17], and by Viale and Weiß in [15]. In particular, this yields new proofs of Weiß's results from [16] and [17], eliminating problems contained in the original proofs.

math.LO↗

arXiv2017

Measurable cardinals and good $Σ_1(κ)$-wellorderings

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.

math.LO↗

arXiv2016

Characterizing large cardinals in terms of layered posets

Given an uncountable regular cardinal $κ$, a partial order is $κ$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $κ$ is stationary in $\mathcal{P}_κ(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $κ$ is weakly compact if and only if every partial order satisfying the $κ$-chain condition is $κ$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $κ$-Knaster partial orders are $κ$-stationarily layered implies that $κ$ is a Mahlo cardinal and every stationary subset of $κ$ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.

math.LO↗

arXiv2013

Free groups and automorphism groups of infinite fields

Let λbe a cardinal with λ=λ^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality λand characteristic p such that the automorphism group of K_0 is a free group of cardinality 2^λand the automorphism group of K_1 is a free abelian group of cardinality 2^λ. This partially answers a question from [8] and complements results from [15], [16] and [17]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that the existence of a cardinal λof uncountable cofinality with the property that there is no field of cardinality λwhose automorphism group is a free group of cardinality greater than λimplies the existence of large cardinals in certain inner models of set theory.

math.LO↗