Mathematics

A question dating to Sibe Mardeši? and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits \mathrm{lim}^n (n>0) of a certain inverse system \mathbf{A} indexed by {^\omega}\omega to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all n -coherent families of functions indexed by {^\omega}\omega to be trivial. In this paper, we prove that, in any forcing extension given by adjoining \beth_\omega -many Cohen reals, \mathrm{lim}^n \mathbf{A} vanishes for all n > 0 . Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional \Delta -system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of \mathrm{lim}^n \mathbf{A} for all n > 0 .

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities, which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via Gödel-McKinsey-Tarski translations.

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending n bits to m≥ n 2 bits violates the dual weak pigeonhole principle: every string y of length m equals the value of the function for some x of length n . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size n d .

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.

Total social welfare relations satisfying Pareto and equity principles on infinite utility streams has revealed a non-constructive nature. In this paper we study more deeply the needed fragment of AC. In particular, we show that such relations need a strictly larger fragment of AC than non-Lebesgue and non-Ramsey sets. We also prove a connection with the Baire property, answering Problem 11.14 posed in "Flutters and chameleon", by Mathias et al.

In 2002 Robert Solovay proved that a subsystem BI of classical second order arithmetic, with bar induction and arithmetical countable choice, can be negatively interpreted in the neutral subsystem BSK of Kleene's intuitionistic analysis FIM using Markov's Principle MP. Combining this result with Kleene's formalized recursive realizability, he established (in primitive recursive arithmetic PRA) that FIM + MP and BI have the same consistency strength. This historical note includes Solovay's original proof, with his permission, and the additional observation that Markov's Principle can be weakened to a double negation shift axiom consistent with Brouwer's creating subject counterexamples.

This is an expository paper about several sophisticated forcing techniques closely related to standard finite support iterations of ccc partial orders. We focus on the four topics of ultrapowers of forcing notions, iterations along templates, Boolean ultrapowers of forcing notions, and restrictions of forcing notions to elementary submodels.

Consider ( κ +++ , κ ++ )↠( κ + ,κ) where κ is an uncountable regular cardinal. By a result of Shelah's we have cof(X∩ κ ++ )=κ for almost all X⊂ κ +++ witnessing this. Here we consider the question if there could be a similar result for X∩ κ + . We use this discussion to give an interesting example of a pseudo Prikry forcing answering a question of Sinapova.

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving elementary embedding from the universe of sets into an inner model, answering a question of Caicedo granting large cardinals. Finally, we show that if \(\kappa\) is strongly compact, then \(V\) is a set forcing extension of the inner model \(\kappa\text{-HOD}\) consisting of sets that are hereditarily ordinal definable from a \(\kappa\)-complete ultrafilter over an ordinal; \(\kappa\text{-HOD}\) seems to be the first nontrivial example of a ground of \(V\) whose definition does not involve forcing.

We prove that, for any natural number n ≥ 1, we can find a finite alphabet Σ and a finitary language L over Σ accepted by a one-counter automaton, such that the ω -power L ∞ := {w 0 w 1. .. ∈ Σ ω | ∀ i ∈ ω w i ∈ L} is Π 0 n-complete. We prove a similar result for the class Σ 0 n .