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We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.
Descriptive set theory and computability theory are closelyrelated fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. In particular, we examine the perfect set property, determinacy, the Baire property, and Lebesgue measurability. We demonstrate that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals; we also show that the Baire property and Lebesgue measurability are both equivalent in complexity to the property of simply being a Borel set, for
$\boldsymbol {\Sigma ^{1}_{2}}$
sets of reals.
In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of firstorder proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as
$\mathsf {P}\neq \mathsf {NP}$
[23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for nonquantified propositional tautologies.
We study the following natural strong variant of destroying Borel ideals:
$\mathbb {P}$
$+$
destroys
$\mathcal {I}$
if
$\mathbb {P}$
adds an
$\mathcal {I}$
positive set which has finite intersection with every
$A\in \mathcal {I}\cap V$
. Also, we discuss the associated variants
of the staruniformity and the starcovering numbers of these ideals.
Among other results, (1) we give a simple combinatorial characterisation when a real forcing
$\mathbb {P}_I$
can
$+$
destroy a Borel ideal
$\mathcal {J}$
; (2) we discuss many classical examples of Borel ideals, their
$+$
destructibility, and cardinal invariants; (3) we show that the Mathias–Prikry,
$\mathbb {M}(\mathcal {I}^*)$
generic real
$+$
destroys
$\mathcal {I}$
iff
$\mathbb {M}(\mathcal {I}^*)\ +$
destroys
$\mathcal {I}$
iff
$\mathcal {I}$
can be
$+$
destroyed iff
$\mathrm {cov}^*(\mathcal {I},+)>\omega $
; (4) we characterise when the Laver–Prikry,
$\mathbb {L}(\mathcal {I}^*)$
generic real
$+$
destroys
$\mathcal {I}$
, and in the case of Pideals, when exactly
$\mathbb {L}(\mathcal {I}^*)$
$+$
destroys
$\mathcal {I}$
; and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.
Let
$\kappa $
be a regular uncountable cardinal, and a cardinal greater than or equal to
$\kappa $
. Revisiting a celebrated result of Shelah, we show that if is close to
$\kappa $
and (= the least size of a cofinal subset of ) is greater than , then can be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that if and , then no
$\kappa $
complete ideal on is weakly saturated.
The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic
${\sf ITL}^{\sf c}_{\Diamond \forall }$
, recently shown to be decidable by FernándezDuque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.
A Cantor series expansion for a real number x with respect to a basic sequence
$Q=(q_1,q_2,\dots )$
, where
$q_i \geq 2$
, is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a
$\boldsymbol {\Pi }^0_3$
complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is
$\boldsymbol {\Pi }^0_3$
complete, and if Q is
$1$
divergent then the sets of normal and ratio normal numbers are
$\boldsymbol {\Pi }^0_3$
complete. We further show that all five nontrivial differences of these sets are
$D_2(\boldsymbol {\Pi }^0_3)$
complete if
$\lim _i q_i=\infty $
and Q is
$1$
divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.
By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if
$M+X\neq \mathbb {R}$
for each meager set M.
A set
$X\subseteq \mathbb {R}$
is meageradditive if
$M+X$
is meager for each meager set M. Recently a theorem on meageradditive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set
$X\subseteq \mathbb {R}$
is meageradditive if and only if it has sharp measure zero, a notion akin to strong measure zero.
We investigate the validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meageradditive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants.
Let
$\mathcal {I}$
be an ideal on
$\omega $
. For
$f,\,g\in \omega ^{\omega }$
we write
$f \leq _{\mathcal {I}} g$
if
$f(n) \leq g(n)$
for all
$n\in \omega \setminus A$
with some
$A\in \mathcal {I}$
. Moreover, we denote
$\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$
(in particular,
$\mathcal {D}_{\mathrm {Fin}}$
denotes the family of all finitetoone functions).
We examine cardinal numbers
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$
and
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$
describing the smallest sizes of unbounded from below with respect to the order
$\leq _{\mathcal {I}}$
sets in
$\mathcal {D}_{\mathrm {Fin}}$
and
$\mathcal {D}_{\mathcal {I}}$
, respectively. For a maximal ideal
$\mathcal {I}$
, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.
We show that
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$
for all ideals
$\mathcal {I}$
with the Baire property and that
$\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$
for all coanalytic weak Pideals (this class contains all
$\bf {\Pi ^0_4}$
ideals). What is more, we give examples of Borel (even
$\bf {\Sigma ^0_2}$
) ideals
$\mathcal {I}$
with
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$
as well as with
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$
.
We also study cardinals
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$
describing the smallest sizes of sets in
$\mathcal {D}_{\mathcal {K}}$
not bounded from below with respect to the preorder
$\leq _{\mathcal {I}}$
by any member of
$\mathcal {D}_{\mathcal {J}}\!$
. Our research is partially motivated by the study of idealQNspaces: those cardinals describe the smallest size of a space which is not idealQN.
In set theory without the Axiom of Choice (
$\mathsf {AC}$
), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinitedimensional subspaces of a given infinitedimensional vector space, as well as the extension of almost disjoint families in infinitedimensional vector spaces to MAD families.
We provide a model theoretical and tree propertylike characterization of
$\lambda $

$\Pi ^1_1$
subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.
We show that if a countable structure M in a finite relational language is not cellular, then there is an agepreserving
$N \supseteq M$
such that
$2^{\aleph _0}$
many structures are biembeddable with N. The proof proceeds by a case division based on mutual algebraicity.
All known structural extensions of the substructural logic
$\textbf{FL}_{\textbf{e}}$
, the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by
$\{\vee , \cdot , 1\}$
equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for noncommutative extensions use an encoding that fails in the presence of commutativity, so andbranching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.
We ask when, for a pair of structures
$\mathcal {A}_1,\mathcal {A}_2$
, there is a uniform effective procedure that, given copies of the two structures, unlabeled, always produces a copy of
$\mathcal {A}_1$
. We give some conditions guaranteeing that there is such a procedure. The conditions might suggest that for the pair of orderings
$\mathcal {A}_1$
of type
$\omega _1^{CK}$
and
$\mathcal {A}_2$
of Harrison type, there should not be any such procedure, but, in fact, there is one. We construct an example for which there is no such procedure. The construction involves forcing. On the way to constructing our example, we prove a general result on modifying Cohen generics.
We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by
$H(F)$
the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in
$H(F)$
, using existential formulas with an arbitrary noncommuting pair of elements as parameters. We show that F is interpreted in
$H(F)$
using computable
$\Sigma _1$
formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of HarrisonTrainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in
$H(F)$
of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in
$H(F)$
. Looking at what was used to arrive at this parameterfree interpretation of F in
$H(F)$
, we give general conditions sufficient to eliminate parameters from interpretations.
A subset of the Cantor cube is nulladditive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are nulladditive, it contains a homeomorphic copy of a set that is not nulladditive, and it has the property
$\unicode{x3b3} $
, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property
$\unicode{x3b3} $
that is not null additive. Settheoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.
We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All ominimal structures on
$(\mathbb {R},<)$
have the property, as do all expansions of
$(\mathbb {R},+,\cdot ,\mathbb {N})$
. Our main analyticgeometric result is that any such expansion of
$(\mathbb {R},<,+)$
by Boolean combinations of open sets (of any arities) either is ominimal or defines an isomorph of
$(\mathbb N,+,\cdot )$
. We also show that any given expansion of
$(\mathbb {R}, <, +,\mathbb {N})$
by subsets of
$\mathbb {N}^n$
(n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.
It is known that every nonuniversal selffull degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using selffull degrees. We show that considerably more can be done by staying entirely inside the collection of nonselffull degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes.
Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any firstorder relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.