arXiv subjects
Publications and source records attributed to Philipp Schlicht.
Kechris, Solecki and Todorčević's $G_0$-dichotomy characterizes Borel graphs that admit a Borel measurable coloring with countably many colors. We show that the analogue to the $G_0$-dichotomy for generalized Cantor spaces fails at the lowest possible complexity, namely for closed graphs.
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings such as Cohen forcing and random algebras. Our approach sidesteps the problem that forcing with the countable chain condition can collapse $ω_1$ by a recent result of Karagila and Schweber. Using this, we show that adding many Cohen reals and random reals leads to different theories. This result is due to Woodin. Thus one can always change the theory of the universe by forcing, just like the continuum hypothesis and its negation can be obtained by forcing over arbitrary models with choice. We further study principles stipulating that the first-order theory of the universe remains the same in all generic extension by a fixed class of forcings. Extending a result of Woodin, we show that even for very restricted classes such as the class of all finite support products of Cohen forcing or the class of all random algebras, this principle implies that all infinite cardinals have countable cofinality.
The open graph dichotomy for a subset $X$ of the Baire space ${}^ωω$ states that any open graph on $X$ either admits a coloring in countably many colors or contains a perfect complete subgraph. This strong version of the open graph axiom for $X$ was introduced by Feng and Todorčević to study definable sets of reals. We first show that its recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup holds for all subsets of the Baire space in Solovay's model, extending a theorem of Feng in dimension $2$. The main theorem lifts this result to generalized Baire spaces ${}^κκ$ in two ways. (1) For any regular infinite cardinal $κ$, the following holds after a Lévy collapse of an inaccessible cardinal $λ>κ$ to $κ^+$. Suppose that $H$ is a $κ$-dimensional box-open directed hypergraph on a subset of ${}^κκ$ such that $H$ is definable from a $κ$-sequence of ordinals. Then either $H$ admits a coloring in $κ$ many colors or there exists a continuous homomorphism from a canonical large directed hypergraph to $H$. (2) If $λ$ is a Mahlo cardinal, then the previous result extends to all box-open directed hypergraphs on any subset of ${}^κκ$ that is definable from a $κ$-sequence of ordinals. This approach is applied to solve several problems about definable subsets of generalized Baire spaces. For instance, we obtain variants of the Hurewicz dichotomy that characterizes subsets of $K_σ$ sets, strong variants of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an $F_σ$ set, the determinacy of Väänänen's perfect set game for all subsets of ${}^κκ$, an asymmetric version of the Baire property and an analogue of the Jayne-Rogers theorem that characterizes $G_δ$-measurable functions.
Let $\Aut(G)$ denote the group of (bi-)continuous automorphisms %and $\Out(G)$ the outer automorphism group of a non-Archimedean Polish group~$G$. We show that for any such $G$ with an invariant countable basis of open subgroups, the group $\Aut(G)$ carries a unique Polish topology that makes its natural action on $G$ continuous. Furthermore, for any class of groups allowing a Borel assignment of such bases, there is a functorial duality to a class of countable groupoids with a meet operation, extending work of the authors with Tent (Coarse groups, and the isomorphism problem for oligomorphic groups, Journal of Mathematical Logic, 2021). This provides an alternative description of the topology of $\Aut(G)$. The results hold for instance for the class of locally Roelcke precompact non-Archimedean groups, which contains most classes studied previously. We further provide a model-theoretic proof that the outer automorphism group $\Out(G)$ of an oligomorphic group $G$ is locally compact, a result due to Paolini and the first author (arXiv:2410.02248).
We study several intertwined hierarchies between $κ$-Ramsey cardinals and measurable cardinals to illuminate the structure of the large cardinal hierarchy in this region. In particular, we study baby versions of measurability introduced by Bovykin and McKenzie and some variants by locating these notions in the large cardinal hierarchy and providing characterisations via filter games. As an application, we determine the theory of the universe up to a measurable cardinal.
In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $κ$-Borel spaces for $κ$ an uncountable (regular) cardinal satisfying $κ^{<κ} = κ$. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from [Coskey-Schlicht 2016, Galeotti 2019, Luecke-Schlicht 2015], and answer some questions contained therein.
We investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness.
Let $κ$ be an uncountable cardinal with $κ=κ^{{<}κ}$. Given a cardinal $μ$, we equip the set ${}^κμ$ consisting of all functions from $κ$ to $μ$ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $κ$. We prove results that allow us to separate several classes of subsets of ${}^κκ$ that consist of continuous images of closed subsets of spaces of the form ${}^κμ$. Important examples of such results are the following: (i) there is a closed subset of ${}^κκ$ that is not a continuous image of ${}^κκ$; (ii) there is an injective continuous image of ${}^κκ$ that is not $κ$-Borel (i.e. that is not contained in the smallest algebra of sets on ${}^κκ$ that contains all open subsets and is closed under $κ$-unions); (iii) the statement "every continuous image of ${}^κκ$ is an injective continuous image of a closed subset of ${}^κκ$" is independent of the axioms of $\mathrm{ZFC}$; and (iv) the axioms of $\mathrm{ZFC}$ do not prove that the assumption "$2^κ>κ^+$'' implies the statement "every closed subset of ${}^κκ$ is a continuous image of ${}^κ(κ^+)$'' or its negation.
Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed subgroups of $S_\infty$. We axiomatise them abstractly as structures with a ternary relation. For appropriate classes of groups, including the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular we can recover an isomorphic copy of~$G$ from $\mathcal M(G)$ in a Borel fashion. A closed subgroup $G$ of $S_\infty$ is called oligomorphic if for each $n$, its natural action on $n$-tuples of natural numbers has only finitely many orbits. We use the concept of a coarse group to show that the isomorphism relation for oligomorphic subgroups of $S_\infty$ is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of $S_\infty$ that are topologically isomorphic to oligomorphic groups.
We introduce and study some variants of a notion of canonical set theoretical truth. By this, we mean truth in a transitive proper class model $M$ of ZFC that is uniquely characterized by some $\in$-formula. We show that there are interesting statements that hold in all such models, but do not follow from ZFC, such as the ground model axiom and the nonexistence of measurable cardinals. We also study a related concept in which we only require M to be fixed up to elementary equivalence. We show that this theory-canonicity also goes beyond provability in ZFC, but it does not rule out measurable cardinals and it does not fix the size of the continuum.
A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the Kechris-Woodin rank and many other ranks in descriptive set theory and real analysis. The complexity of many natural ranks is $Π^1_1$ or $Σ^1_2$. We propose to understand the least length of ranks on a set as a measure of its complexity. Therefore, the aim is to understand which lengths such ranks may have. The main result determines the suprema of lengths of countable ranks at the first and second projective levels. Furthermore, we characterise the existence of countable ranks on specific classes of $Σ^1_2$ sets. The connections arising between $Σ^1_2$ sets with countable ranks on the one hand and $Σ^1_2$ Borel sets on the other lead to a conjecture that unifies several results in descriptive set theory such as the Mansfield-Solovay theorem and a recent result of Kanovei and Lyubetsky.
Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a given ideal. We prove that for all $σ$-ideals $I$ such that the ideal forcing $\mathbb{P}_I$ of Borel sets modulo $I$ is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas with respect to $\mathbb{P}_I$ that we call internal absoluteness. In addition, we show that it is equivalent to measurability with respect to $I$ together with $1$-step absoluteness for the poset $\mathbb{P}_I$. These equivalences are new even for Cohen and random forcing and they are, to the best of our knowledge, the first precise equivalences between regularity and absoluteness beyond the second level of the projective hierarchy.
The decision time of an infinite time algorithm is the supremum of its halting times over all real inputs. The decision time of a set of reals is the least decision time of an algorithm that decides the set; semidecision times of semidecidable sets are defined similary. It is not hard to see that $ω_1$ is the maximal decision time of sets of reals. Our main results determine the supremum of countable decision times as $σ$ and that of countable semidecision times as $τ$, where $σ$ and $τ$ denote the suprema of $Σ_1$- and $Σ_2$-definable ordinals, respectively, over $L_{ω_1}$. We further compute analogous suprema for singletons.
We investigate generalizations of the topology of the higher Cantor space on $2^κ$, based on arbitrary ideals rather than the bounded ideal on $κ$. Our main focus is on the topology induced by the nonstationary ideal, and we call this topology the nonstationary topology, or also the Edinburgh topology on $2^κ$. It may be of independent interest that as a side result, we show $κ$-Silver forcing to satisfy a strong form of Axiom $A$ not only if $κ$ is inaccessible (which is well-known), but also under the assumption $\diamondsuit_κ$.
We study principles of the form: if a name $σ$ is forced to have a certain property $φ$, then there is a ground model filter $g$ such that $σ^g$ satisfies $φ$. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are: Any forcing axiom can be expressed as a name principle. For instance, $\mathsf{PFA}$ is equivalent to a principle for rank $1$ names (equivalently, nice names) for subsets of $ω_1$, and a principle for rank $2$ names for sets of reals. Moreover, $λ$-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of $\mathsf{BFA}$ via generic absoluteness is a corollary. We further systematically study name principles where $φ$ is a notion of largeness for subsets of $ω_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.
We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.
We use generalizations of concepts from descriptive set theory to study combinatorial objects of uncountable regular cardinality, focussing on higher Kurepa trees and the representation of the sets of cofinal branches through such trees as continuous images of function spaces. For different types of uncountable regular cardinals $κ$, our results provide a complete picture of all consistent scenarios for the representation of sets of cofinal branches through $κ$-Kurepa trees as retracts of the generalized Baire space ${}^κκ$ of $κ$. In addition, these results can be used to determine the consistency of most of the corresponding statements for continuous images of ${}^κκ$.
The present work looks at semiautomatic rings with automatic addition and comparisons which are dense subrings of the real numbers and asks how these can be used to represent geometric objects such that certain operations and transformations are automatic. The underlying ring has always to be a countable dense subring of the real numbers and additions and comparisons and multiplications with constants need to be automatic. It is shown that the ring can be selected such that equilateral triangles can be represented and rotations by 30 degrees are possible, while the standard representation of the b-adic rationals does not allow this.