arXiv subjects

Philipp Schlicht

Publications and source records attributed to Philipp Schlicht.

At least 37 records · Page 2Linked to original sources

arXiv2021

Borel subsets of the real line and continuous reducibility

We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order $ω_1$, its reverse $ω_1^\star$ and the poset $\mathcal{P}(ω)/\mathsf{fin}$ of inclusion modulo finite error. As a consequence of its proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to $\mathbb{Q}$. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In our second main theorem, we introduce a combinatorial property that is shown to characterize those $F_σ$ sets that are reducible to $\mathbb{Q}$. This is applied to construct a minimal set below $\mathbb{Q}$ and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.

math.LO↗

arXiv2021

The exact strength of the class forcing theorem

The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the corresponding forcing extensions are forced and forced statements are true -- is equivalent over Gödel-Bernays set theory GBC to the principle of elementary transfinite recursion $\text{ETR}_{\text{Ord}}$ for class recursions of length $\text{Ord}$. It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal{L}_{\text{Ord},ω}(\in,A)$, allowing any class parameter $A$; to the existence of truth predicates for the language $\mathcal{L}_{\text{Ord},\text{Ord}}(\in,A)$; to the existence of $\text{Ord}$-iterated truth predicates for first-order set theory $\mathcal{L}_{ω,ω}(\in,A)$; to the assertion that every separative class partial order $\mathbb{P}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text{Ord}+1$. Unlike set forcing, if every class forcing notion $\mathbb{P}$ has a forcing relation merely for atomic formulas, then every such $\mathbb{P}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between GBC and Kelley-Morse set theory KM.

math.LO↗

arXiv2021

Lebesgue's Density Theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.

math.LO↗

arXiv2021

Long Games and $σ$-Projective Sets

We prove a number of results on the determinacy of $σ$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $σ$-projective determinacy and the determinacy of certain classes of games of variable length ${<}ω^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $σ$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $σ$-projective games of a given countable length and of games with payoff in the smallest $σ$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).

math.LO↗

arXiv2020

How to have more things by forgetting how to count them

Cohen's first model is a model of Zermelo--Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal $κ$. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding $κ$ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set $A$ which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of "Adding a Cohen subset" by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, $2^A$ is extremally disconnected, or $[A]^{<ω}$ is Dedekind-finite.

math.LO↗

arXiv2020

Reachability for infinite time Turing machines with long tapes

Infinite time Turing machine models with tape length $α$, denoted $T_α$, strengthen the machines of Hamkins and Kidder [HL00] with tape length $ω$. A new phenomenon is that for some countable ordinals $α$, some cells cannot be halting positions of $T_α$ given trivial input. The main open question in [Rin14] asks about the size of the least such ordinal $δ$. We answer this by providing various characterizations. For instance, $δ$ is the least ordinal with any of the following properties: (a) For some $ξ<α$, there is a $T_ξ$-writable but not $T_α$-writable subset of $ω$. (b) There is a gap in the $T_α$-writable ordinals. (c) $α$ is uncountable in $L_{λ_α}$. Here $λ_α$ denotes the supremum of $T_α$-writable ordinals, i.e. those with a $T_α$-writable code of length $α$. We further use the above characterizations, and an analogue to Welch's submodel characterization of the ordinals $λ$, $ζ$ and $Σ$, to show that $δ$ is large in the sense that it is a closure point of the function $α\mapsto Σ_α$, where $Σ_α$ denotes the supremum of the $T_α$-accidentally writable ordinals.

math.LO↗

arXiv2019

The isomorphism problem for tree-automatic ordinals with addition

This paper studies tree-automatic ordinals (or equivalently, well-founded linearly ordered sets) together with the ordinal addition operation +. Informally, these are ordinals such that their elements are coded by finite trees for which the linear order relation of the ordinal and the ordinal addition operation can be determined by tree automata. We describe an algorithm that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic.

cs.FL↗

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

Characterizations of pretameness and the Ord-cc

It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of $\mathsf{ZF}^-$, that is $\mathsf{ZF}$ without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of $\mathsf{ZF}$. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the $\mathrm{Ord}$-chain condition.

math.LO↗

arXiv2017

Sufficient conditions for the forcing theorem, and turning proper classes into sets

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself), including the three properties presented in this paper, imply yet another regularity property for class forcing notions, namely that proper classes of the ground model cannot become sets in a generic extension, that is they do not have set-sized names in the ground model. We then show that over certain models of Gödel-Bernays set theory without the power set axiom, there is a notion of class forcing which turns a proper class into a set, however does not satisfy the forcing theorem. Moreover, we show that the property of not turning proper classes into sets can be used to characterize pretameness over such models of Gödel-Bernays set theory.

math.LO↗

arXiv2017

A hierarchy of Ramsey-like cardinals

We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy.

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

Continuous reducibility and dimension of metric spaces

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,τ)$ is called the \emph{Wadge quasi-order} for $(X,τ)$. We further show that this quasi-order, restricted to the Borel subsets of a Polish space $(X,τ)$, is a \emph{well-quasiorder (wqo)} if and only if $(X,τ)$ has dimension $0$, as an application of the main result. Moreover, we give further examples of applications of the technique, which is based on a construction of graph colorings.

math.LO↗

arXiv2017

Perfect subsets of generalized Baire spaces and long games

We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space ${}^λλ$, where $λ$ is an uncountable cardinal with $λ^{<λ}=λ$. In the first main theorem, we show that that the perfect set property for all subsets of ${}^λλ$ that are definable from elements of ${}^λ\mathrm{Ord}$ is consistent relative to the existence of an inaccessible cardinal above $λ$. In the second main theorem, we introduce a Banach-Mazur type game of length $λ$ and show that the determinacy of this game, for all subsets of ${}^λλ$ that are definable from elements of ${}^λ\mathrm{Ord}$ as winning conditions, is consistent relative to the existence of an inaccessible cardinal above $λ$. We further obtain some related results about definable functions on ${}^λλ$ and consequences of resurrection axioms for definable subsets of ${}^λλ$.

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

Randomness via infinite computation and effective descriptive set theory

We study randomness beyond $Π^1_1$-randomness and its Martin-Löf type variant, introduced in \cite{MR2340241} and further studied in \cite{Continuous-higher-randomness}. The class given by the infinite time Turing machines (\ITTM s), introduced by Hamkins and Kidder, is strictly between $Π^1_1$ and $Σ^1_2$. We prove that the natural randomness notions associated to this class have several desirable properties resembling those of the classical random notions such as Martin-Löf randomness, and randomness notions defined via effective descriptive set theory such as $Π^1_1$-randomness. For instance, mutual randoms do not share information and can be characterized as in van Lambalgen's theorem. We also obtain some differences to the hyperarithmetic setting. Already at the level of $Σ^1_2$, some properties of randomness notions are independent \cite{Infinite-computations}. Towards the results about randomness, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool, we prove facts of independent interest about random forcing over admissible sets and increasing unions of admissible sets. These results are also useful for more efficient proofs of some classical results about hyperarithmetic sets.

math.LO↗

arXiv2016

The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_σ$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^ωω$. We consider the analogous statement (which we call Hurewicz dichotomy) for $Σ^1_1$ subsets of the generalized Baire space ${}^κκ$ for a given uncountable cardinal $κ$ with $κ=κ^{<κ}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal preserving class-forcing extension in which the Hurewicz dichotomy for $Σ^1_1$ subsets of ${}^κκ$ holds at all uncountable regular cardinals $κ$, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for $Σ^1_1$ sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the $κ$-perfect set property, the $κ$-Miller measurability, and the $κ$-Sacks measurability.

math.LO↗

arXiv2015

Recognizable sets and Woodin cardinals: Computation beyond the constructible universe

We call a subset of an ordinal $λ$ recognizable if it is the unique subset $x$ of $λ$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $λ$ as input, halts with the final state $0$. Equivalently, such a set is the unique subset $x$ which satisfies a given $Σ_1$ formula in $L[x]$. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals. (1) Computable sets are elements of $L$, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals. (2) A subset of a countable ordinal $λ$ is in the recognizable closure for subsets of $λ$ if and only if it is an element of $M^{\infty}$, where $M^{\infty}$ denotes the inner model obtained by iterating the least measure of $M_1$ through the ordinals, and where the recognizable closure for subsets of $λ$ is defined by closing under relative recognizability for subsets of $λ$.

math.LO↗