The Principle of Open Induction on Cantor space and the Approximate-Fan Theorem

The paper is a contribution to intuitionistic reverse mathematics. We work in a weak formal system for intuitionistic analysis. The Principle of Open Induction on Cantor space is the statement that every open subset of Cantor space that is progressive with respect to the lexicographical ordering of Cantor space coincides with Cantor space. The Approximate-Fan Theorem is an extension of the Fan Theorem that follows from Brouwer's principle of induction on bars in Baire space and implies the Principle of Open Induction on Cantor space. The Principle of Open Induction in Cantor space implies the Fan Theorem, but, conversely the Fan Theorem does not prove the Principle of Open Induction on Cantor space. We list a number of equivalents of the Principle of Open Induction on Cantor space and also a number of equivalents of the Approximate-Fan Theorem

Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative

The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic Mathematics BIM, and then search for statements that are, over BIM, equivalent to Brouwer's Fan Theorem or to its positive denial, Kleene's Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionistic interpretation and Kleene's Alternative is true in the model of BIM consisting of the Turing-computable functions. The task of finding equivalents of Kleene's Alternative is, intuitionistically, a nontrivial extension of finding equivalents of the Fan Theorem, although there is a certain symmetry in the arguments that we shall try to make transparent. We introduce closed-and-separable subsets of Baire space and of the set of the real numbers. Such sets may be compact and also positively noncompact. The Fan Theorem is the statement that Cantor space, or, equivalently, the unit interval, is compact, and Kleene's Alternative is the statement that Cantor space, or, equivalently, the unit interval is positively noncompact. The class of the compact closed-and-separable sets and also the class of the closed-and-separable sets that are positively noncompact are characterized in many different ways and a host of equivalents of both the Fan Theorem and Kleene's Alternative is found

The Fan Theorem, its strong negation, and the determinacy of games

IIn the context of a weak formal theory called Basic Intuitionistic Mathematics $\mathsf{BIM}$, we study Brouwer's Fan Theorem and a strong negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene's Alternative is equivalent to strong negations of these statements. We also discuss finite and infinite games and introduce a constructively useful notion of determinacy. We prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem, saying that every subset of Cantor space is, in our constructively meaningful sense, determinate, and show that Kleene's Alternative is equivalent to a strong negation of a special case of this theorem. We then consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic, and prove that the Fan Theorem is equivalent to each of these theorems and that Kleene's Alternative is equivalent to strong negations of them. We end with a note on a possibly important statement, provable from principles accepted by Brouwer, that one might call a Strong Fan Theorem.Comment: arXiv admin note: text overlap with arXiv:1106.273

Brouwer’s Real Thesis on Bars

L.E.J. Brouwer made a mistake in the formulation of his famous bar theorem, as was pointed out by S.C. Kleene. By repeating this mistake several times, Brouwer has caused confusion. We consider the assumption underlying his bar theorem, calling it Brouwer’s Thesis. This assumption is not refuted by Kleene’s example and we use it to obtain a conclusion different from Brouwer’s. Thus we come to support a view first expressed and defended by E. Martino and P. Giaretta in [Martino 1981]. We also indicate that Brouwer’s Thesis has many more applications than Brouwer dreamt of

Health risks by bromomethane and other toxic gases in import cargo ship containers

Containers are increasingly used for the worldwide transport of all kinds of goods. Consistent with national and international regulations on pest controls, a growing proportion of these containers undergoes fumigation. Frequently, the prescribed labelling is missing. According to literature, this situation may lead to accidents and represents a significant health risk to dock workers, inspectors and custom workers. Furthermore, warehouse workers and even consumers may come in contact with these toxic fumigants. Presented measurement data underline this health risks due to bromomethane but also due to other fumigants and, surprisingly, due to further noxious gases. So far, no routine method for sensitive and specific measurements on the spot has been available. The consequences of container fumigation should always be carefully weighed up, and alternatives to pesticides, e.g. heat treatment or atmospheres with reduced oxygen and for high CO2 concentrations should be considered. In addition, stringent international controls as well as sanctions if IMO’s “Recommendations on the safe use of pesticides in ships” are disregarded are required

A global genomic analysis of Salmonella Concord reveals lineages with high antimicrobial resistance in Ethiopia.

Antimicrobial resistant Salmonella enterica serovar Concord (S. Concord) is known to cause severe gastrointestinal and bloodstream infections in patients from Ethiopia and Ethiopian adoptees, and occasional records exist of S. Concord linked to other countries. The evolution and geographical distribution of S. Concord remained unclear. Here, we provide a genomic overview of the population structure and antimicrobial resistance (AMR) of S. Concord by analysing genomes from 284 historical and contemporary isolates obtained between 1944 and 2022 across the globe. We demonstrate that S. Concord is a polyphyletic serovar distributed among three Salmonella super-lineages. Super-lineage A is composed of eight S. Concord lineages, of which four are associated with multiple countries and low levels of AMR. Other lineages are restricted to Ethiopia and horizontally acquired resistance to most antimicrobials used for treating invasive Salmonella infections in low- and middle-income countries. By reconstructing complete genomes for 10 representative strains, we demonstrate the presence of AMR markers integrated in structurally diverse IncHI2 and IncA/C2 plasmids, and/or the chromosome. Molecular surveillance of pathogens such as S. Concord supports the understanding of AMR and the multi-sector response to the global AMR threat. This study provides a comprehensive baseline data set essential for future molecular surveillance