Mathematics

We prove that the automorphism group of the Urysohn diversity is a universal Polish group. Furthermore we show that the automorphism group of the rational Urysohn diversity has ample generics, a dense conjugacy class and that it embeds densely into the automorphism group of the (full) Urysohn diversity. It follows that this latter group also has a dense conjugacy class.

Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of Complete Ordered Fields with which Real Analysis starts. Groups abound in mathematical sciences, while by Dedekind's theorem there exists only one complete ordered field, up to isomorphism. Cayley's theorem in Abstract Algebra implies that the axioms of group theory completely axiomatize the class of permutation sets that are closed under composition and inversion. In this article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers.

In this paper we consider the modal logic with both Box and Diamond arising fromKripke models with a crisp accessibility and whose propositions are valued over the stan-dard Godel algebra [0,1]G. We provide an axiomatic system extending the one from [3]for models with a valued accessibility with Dunn axiom from positive modal logics, andshow it is strongly complete with respect to the intended semantics. The axiomatizationsof the most usual frame restrictions are given too. We also prove that in the studied logicit is not possible to get Box as an abbreviation of Diamond, nor vice-versa, showing that indeedthe axiomatic system we present does not coincide with any ofthe mono-modal fragmentspreviously axiomatized in the literature.

We give axioms in the language of rings augmented by a 1-ary predicate symbol Fin(x) with intended interpretation in the Boolean algebra of idempotents as the ideal of finite elements, i.e. finite unions of atoms. We prove that any commutative unital ring satisfying these axioms is elementarily equivalent to a restricted product of connected rings. This is an extension of the results in \cite{elem-prod} for products. While the results in \cite{elem-prod} give a converse to the Feferman-Vaught theorem for products, our results prove the same for restricted products. We give a complete set of axioms in the language of rings for the ring of adeles of a number field, uniformly in the number field.

Let M be a λ -indexed (that is, Jensen indexed) premouse. We prove that M is iterable with respect to standard λ -iteration rules iff M is iterable with respect to a natural version of Mitchell-Steel iteration rules. Using this equivalence, we describe a background construction for λ -indexed mice, analogous to traditional background constructions for Mitchell-Steel indexed mice, and which absorbs Woodin cardinals from the background universe. We also prove some facts regarding the correspondence between standard iteration trees and u-iteration trees on premice with Mitchell-Steel indexing.

Alternative set theory (AST) may be suitable for the ones who try to capture objects or phenomenons with some kind of indefiniteness of a border. While AST provides various notions for advanced mathematical studies, correspondence of them to that of conventional ones are not fully developed. This paper presents basic topological concepts in AST, and shows their correspondence with those of conventional ones, and isomorphicity of a system of real numbers in AST to that of conventional ones.

One can consider μ -Martin-Löf randomness for a probability measure μ on 2 ω , such as the Bernoulli measure μ p given p∈(0,1) . We study Bernoulli randomness of sequences in n ω with parameters p 0 , p 1 ,…, p n−1 , and we introduce a biased version of normality. We prove that every Bernoulli random real is normal in the biased sense, and this has the corollary that the set of biased normal reals has full Bernoulli measure in n ω . We give an algorithm for computing biased normal sequences from normal sequences, so that we can give explicit examples of biased normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and biased normality.

The Beth Definability Property holds for an algebraizable logic if and only if every epimorphism in the corresponding category of algebras is surjective. Using this technique, Urquhart in 1999 showed that the Beth Definability Property fails for a wide class of relevant logics, including T, E, and R. However, the counterexample for those logics does not extend to the algebraic counterpart of the super relevant logic KR, the so-called Boolean monoids. Following a suggestion of Urquhart, we use modular lattices constructed by Freese to show that epimorphisms need not be surjective in a wide class of relation algebras. This class includes the Boolean monoids, and thus the Beth Definability Property fails for KR.

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨ Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨ Δ , stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢ Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence associated with ⊢ Δ coincides with ⊨ Δ .

Any binary string can be associated with a unary predicate P on N . In this paper we investigate subsets named by a predicate P such that the relation P(x+y) has finite VC dimension. This provides a measure of complexity for binary strings with different properties than the standard string complexity function (based on diversity of substrings). We prove that strings of bounded VC dimension are meagre in the topology of the reals, provide simple rules for bounding the VC dimension of a string, and show that the bi-infinite strings of VC dimension d are a non-sofic shift space. Additionally we characterize the irreducible strings of low VC dimension (0,1 and 2), and provide connections to mathematical logic.