Mathematics

For given Boolean algebras A and B we endow the space H(A,B) of all Boolean homomorphisms from A to B with various topologies and study convergence properties of sequences in H(A,B) . We are in particular interested in the situation when B is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on H(A,B) for a Boolean algebra B carrying a strictly positive measure and convergence properties of sequences of measures on A .

We work with symmetric inner models of forcing extensions based on strongly compact Prikry forcing to extend some known results.

The main result of this paper is to show that, if κ is the smallest real-valued measurable cardinal not greater than 2 ℵ 0 , then there exists a complete metric space of cardinality not greater than 2 κ admitting a Kuratowski partition.

We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field. Moreover, we prove that any \emph{absolute} amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Spinas' and Hein's conjecture that $\add(\ideal{I}_\spl) \leq \mathfrak{b}$.

Euclid uses an undefined notion of "equal figures", to which he applies the common notions about equals added to equals or subtracted from equals. When (in previous work) we formalized Euclid Book~I for computer proof-checking, we had to add fifteen axioms about undefined relations "equal triangles" and "equal quadrilaterals" to replace Euclid's use of the common notions. In this paper, we offer definitions of "equal triangles" and "equal quadrilaterals", that Euclid could have given, and prove that they have the required properties. This removes the need for adding new axioms. The proof uses the theory of proportions. Hence we also discuss the "early theory of proportions", which has a long history.

This work presents a generalized notion of multiset mapping thus resolving a long standing obstacle in structural study of multiset processing. It has been shown that the mapping defined herein can model a vast array of notions as special cases and also handels diverse situations in multiset rewriting transformations. Specifically, this paper unifies and generalizes the works of Parikh(1966), Hickman(1980), Khomenko(2003) and Nazmul(2013).

There is a longstanding debate in the logico-philosophical community as to why the Gödelian sentences of a consistent and sufficiently strong theory are true. The prevalent argument seems to be something like this: since every one of the Gödelian sentences of such a theory is equivalent to the theory's consistency statement, even provably so inside the theory, the truth of those sentences follows from the consistency of the theory in question. So, Gödelian sentences of consistent theories should be true. In this paper, we show that Gödelian sentences of only sound theories are true; and there is a long road from consistency to soundness, indeed a hierarchy of conditions which are satisfied by some theories and falsified by others. We also study the truth of Rosserian sentences and provide necessary and sufficient conditions for the truth of Rosserian (and also Gödelian) sentences of theories.

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author which establishes the consistency, relative to a weakly compact cardinal, of lim s A=0 for all s≥1 for a certain pro-abelian group A ; we show that that work's arguments carry implications for the vanishing and additivity of the lim s functors over a substantially more general class of pro-abelian groups indexed by N N .

In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers n≥3 , R cl (ω+n,3) R cl (ω+n,3) ≥ ω 2 ⋅n+ω⋅(R(n,3)−n)+n ≤ ω 2 ⋅n+ω⋅(R(2n−3,3)+1)+1 where R cl (⋅,⋅) and R(⋅,⋅) denote the closed Ramsey numbers and the classical Ramsey numbers respectively. We also establish the following asymptotically weaker upper bound R cl (ω+n,3)≤ ω 2 ⋅n+ω⋅( n 2 −4)+1 eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.

In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems: whether a generated class is a real closed field and whether there exists a computable presentation of a generated class. We prove a series of theorems that lead to the result that there are no computable presentations neither for polynomial time computable no even for E n -computable real numbers, where E n is a level in Grzegorczyk hierarchy, n≥2 . We also propose a criterion of computable presentability of an archimedean ordered field.