Mathematics

Generalizing and simplifying recent work of Dobrinen, we show that if L is a finite binary relational language and F is a finite set of finite irreducible L -structures, then the class K=Forb(F) has finite big Ramsey degrees.

We study a pair of weakenings of the classical partition relation ν→(μ ) 2 λ recently introduced by Bergfalk-Hrušák-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on ν -many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complementary positive and negative results indicating the effect of large cardinals, forcing axioms, and square principles on these partition relations. We also prove a consistency result indicating that a non-trivial instance of the stronger of these two partition relations can hold at the continuum.

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we generalise this result to \kappa-prime models, for \kappa a regular uncountable cardinal or \aleph_\epsilon.

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was also addressed by Shelah ([11, Problem 0.5]) and concerns the definition of a forcing which is κ k appa -bounding, <κ -closed and κ + -cc, for κ inaccessible.

We show that it is provable in PA that there is an arithmetically definable sequence { ϕ n :n∈ω} of Π 0 2 -sentences, such that - PRA+ { ϕ n :n∈ω} is Π 0 2 -sound and Π 0 1 -complete - the length of ϕ n is bounded above by a polynomial function of n with positive leading coefficient - PRA+ ϕ n+1 always proves 1-consistency of PRA+ ϕ n . One has that the growth in logical strength is in some sense "as fast as possible", manifested in the fact that the total general recursive functions whose totality is asserted by the true Π 0 2 -sentences in the sequence are cofinal growth-rate-wise in the set of all total general recursive functions. We then develop an argument which makes use of a sequence of sentences constructed by an application of the diagonal lemma, which are generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth in Mathematics". The argument establishes the result that it is provable in PA that P≠NP . We indicate how to pull the argument all the way down into EFA.

In this paper, we shall show that the following translation I M from the propositional fragment L 1 of Leśniewski's ontology to modal logic KTB is sound: for any formula ϕ and ψ of L 1 , it is defined as (M1) I M (ϕ∨ψ) = I M (ϕ)∨ I M (ψ), (M2) I M (¬ϕ) = ¬ I M (ϕ), (M3) I M (ϵab) = ◊ p a ⊃ p a .∧.□ p a ⊃□ p b .∧.◊ p b ⊃ p a , where p a and p b are propositional variables corresponding to the name variables a and b , respectively. We shall give some comments including some open problems and my conjectures.

Adapting standard methods from geometric measure theory, we provide an example of a polynomial-valued measure μ on tame sets in R d which satisfies many desirable properties. Among these is strict monotonicity: the measure of a proper subset is strictly less than the measure of the whole set. Using techniques from non-standard analysis, we display that the domain of μ can be extended to all subsets of R d (up to equivalence modulo infinitesimals). The resulting extension is a numerosity function that encodes the i -dimensional Hausdorff measure for all i∈N , as well as the i -th intrinsic volume functions.

Goodstein's principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to grow very quickly, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we explore notions of optimality for such notation systems and apply them to the classical Goodstein process, to a weaker variant based on multiplication rather than exponentiation, and to a stronger variant based on the Ackermann function. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein's result.

We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let M=( V M ,P) be a countable model of second order set theory ZF 2 (with universe V M and classes P ) which models " κ is super Reinhardt". We show that there are unboundedly many μ<κ such that there is j such that ( V M ,j) models ZF(j)+ " μ is Reinhardt, as witnessed by j ". In particular, j↾X∈ V M for all X∈ V M (but we allow j∉P ).

It is well known from universal algebra that, for every signature Σ , there exist algebras over Σ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Σ -algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Σ -algebras to Set , has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U , from the category of Σ -multialgebras to Set , does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), it corresponds a unique homomorphism, which ressembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint.