Transfinite inductions producing coanalytic sets

  title={Transfinite inductions producing coanalytic sets},
  author={Zolt{\'a}n Vidny{\'a}nszky},
  journal={Fundamenta Mathematicae},
A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that in $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set. 
On the scope of the Effros theorem
All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group G is Effros (that is, every continuous transitive action of G on a non-meager spaceExpand
Beyond Erdős-Kunen-Mauldin: Shift-compactness properties and singular sets
Abstract The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of R which is Baire (= has the Baire property, BP) or measurable is shift-compact: it contains some subsequence ofExpand
Maximal discrete sets
We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, I-mad families, maximalExpand
Tree forcing and definable maximal independent sets in hypergraphs
We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbf{\Delta}^1_2$Expand
Definable MAD families and forcing axioms
We show that under the Bounded Proper Forcing Axiom and an anti-large cardinal assumption, there is a $\mathbf{\Pi}^1_2$ MAD family.
ICLE Set theory and the analyst
This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure—category-measure duality and non-duality, as it were. The bulkExpand
Every zero-dimensional homogeneous space is strongly homogeneous under determinacy
It is shown that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. Expand
Set theory and the analyst
This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure—category-measure duality and non-duality, as it were. The bulkExpand
All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. Inspired by this theorem, we obtain the following results: ‚ AssumingExpand


Analytic and coanalytic families of almost disjoint functions
Abstract If is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable , no member of which is covered by finitely manyExpand
Infinite Combinatorics and Definability
It is shown that there cannot be a Borel subset of [ω] ω which is a maximal independent family, and it is consistent that any ω 2 cover of reals by Borel sets has an ω 1 subcover. Expand
Definable sets of generators in maximal cofinitary groups
Abstract A group G ⩽ Sym ( N ) is cofinitary if g has finitely many fixed points for every g ∈ G except the identity element. In this paper, we discuss the definability of maximal cofinitary groupsExpand
A Pi ^1_1-uniformization principle for reals
We introduce a Π 1 1 -uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω 1 ) L = ω 1 . This principle is then applied to derive the equivalence, to suitableExpand
A co-analytic maximal set of orthogonal measures
Abstract We prove that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of PreissExpand
Higher recursion theory
Hyperarithmetic theory is the first step beyond classical recursion theory. It is the primary source of ideas and examples in higher recursion theory. It is also a crossroad for several areas ofExpand
Arcs in the plane
Abstract Assuming PFA, every uncountable subset E of the plane meets some C 1 arc in an uncountable set. This is not provable from MA ( ℵ 1 ) , although in the case that E is analytic, this is a ZFCExpand
Some additive properties of sets of real numbers
Some problems concerning the additive properties of subsets of R are investigated. From a result of G. G . Lorentz in additive number theory, we show that if P is a nonempty perfect subset of R, thenExpand
Descriptive Set Theory
Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitraryExpand
The theory of countable analytical sets
The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy,Expand