Mathematics

This paper discusses the scope and significance of the so-called triviality result stated by Allan Gibbard for indicative conditionals, showing that if a conditional operator satisfies the Law of Import-Export, is supraclassical, and is stronger than the material conditional, then it must collapse to the material conditional. Gibbard's result is taken to pose a dilemma for a truth-functional account of indicative conditionals: give up Import-Export, or embrace the two-valued analysis. We show that this dilemma can be averted in trivalent logics of the conditional based on Reichenbach and de Finetti's idea that a conditional with a false antecedent is undefined. Import-Export and truth-functionality hold without triviality in such logics. We unravel some implicit assumptions in Gibbard's proof, and discuss a recent generalization of Gibbard's result due to Branden Fitelson.

The H -coloring problem for undirected simple graphs is a computational problem from a huge class of the constraint satisfaction problems (CSP): an H -coloring of a graph G is just a homomorphism from G to H and the problem is to decide for fixed H , given G , if a homomorphism exists or not. The dichotomy theorem for the H -coloring problem was proved by Hell and Nešetřil in 1990 (an analogous theorem for all CSPs was recently proved by Zhuk and Bulatov) and it says that for each H the problem is either p -time decidable or NP -complete. Since negations of unsatisfiable instances of CSP can be expressed as propositional tautologies, it seems to be natural to investigate the proof complexity of CSP. We show that the decision algorithm in the p -time case of the H -coloring problem can be formalized in a relatively weak theory and that the tautologies expressing the negative instances for such H have short proofs in propositional proof system R ∗ (log) , a mild extension of resolution. In fact, when the formulas are expressed as unsatisfiable sets of clauses they have p -size resolution proofs. To establish this we use a well-known connection between theories of bounded arithmetic and propositional proof systems. We complement this result by a lower bound result that holds for many weak proof systems for a special example of NP -complete case of the H -coloring problem, using the known results about proof complexity of the Pigeonhole Principle.

We study H-structures associated to SU-rank one measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

We analyze the hereditarily ordinal definable sets HOD in M n (x)[g] for a Turing cone of reals x , where M n (x) is the canonical inner model with n Woodin cardinals build over x and g is generic over M n (x) for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming Π 1 n+2 -determinacy, for a Turing cone of reals x , HOD M n (x)[g] = M n ( M ∞ | κ ∞ ,Λ), where M ∞ is a direct limit of iterates of M n+1 , δ ∞ is the least Woodin cardinal in M ∞ , κ ∞ is the least inaccessible cardinal in M ∞ above δ ∞ , and Λ is a partial iteration strategy for M ∞ . It will also be shown that under the same hypothesis HOD M n (x)[g] satisfies GCH .

We show that every non-Haar-null analytic subset of Z ω contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of Z ω form a Δ 1 2 set.

The Hausdorff δ -dimension game was introduced by Das, Fishman, Simmons and {Urba{ń}ski} and shown to characterize sets in R d having Hausdorff dimension ≤δ . We introduce a variation of this game which also characterizes Hausdorff dimension and for which we are able to prove an unfolding result similar to the basic unfolding property for the Banach-Mazur game for category. We use this to derive a number of consequences for Hausdorff dimension. We show that under AD any wellordered union of sets each of which has Hausdorff dimension ≤δ has dimension ≤δ . We establish a continuous uniformization result for Hausdorff dimension. The unfolded game also provides a new proof that every Σ 1 1 set of Hausdorff dimension ≥δ contains a compact subset of dimension ≥ δ ′ for any δ ′ <δ , and this result generalizes to arbitrary sets under AD .

We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of L κ + , κ + -indescribability and ? 1 ξ -indescribability of a cardinal κ for all ξ< κ + . In this context, universal ? 1 ξ formulas exist, there is a normal ideal associated to ? 1 ξ -indescribability and the notions of ? 1 ξ -indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given a regular cardinal μ , we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's \cite{MR3894041} sequence langle ? ξ :ξ<μ??of derived topologies on μ to ??? ξ :ξ< μ + ??. Finally, we prove that for all ξ< μ + , if there is a stationary set of α<μ that have a high enough degree of indescribability, then there are stationarily-many α<μ that are nonisolated points in the space (μ, ? ξ+1 ) .

We investigate higher-dimensional Δ -systems, isolating a particular definition thereof and proving a higher-dimensional version of the classical Δ -system lemma. We then present two applications of this lemma to problems involving the interplay between forcing and partition relations involving the reals.

We relate the decidability problem for BS with unordered cartesian product with Hilbert's Tenth problem and prove that BS with unordered cartesian product is NP-complete.

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.