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"Kossak, Roman"
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THE LATTICE PROBLEM FOR MODELS OF$\\mathsf {PA}
The lattice problem for models of Peano Arithmetic ($\\mathsf {PA}$) is to determine which lattices can be represented as lattices of elementary submodels of a model of$\\mathsf {PA}$, or, in greater generality, for a given model$\\mathcal {M}$, which lattices can be represented as interstructure lattices of elementary submodels$\\mathcal {K}$of an elementary extension$\\mathcal {N}$such that$\\mathcal {M}\\preccurlyeq \\mathcal {K}\\preccurlyeq \\mathcal {N}$. The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.
Journal Article
Logic Without Borders
In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy and computer science continue to be important and noticeable. The series publishes conference proceedings as well as monographs written by leading researchers in mathematical logic.
eBook
Set Theory, Arithmetic, and Foundations of Mathematics
This collection of papers from various areas of mathematical logic showcases the remarkable breadth and richness of the field. Leading authors reveal how contemporary technical results touch upon foundational questions about the nature of mathematics. Highlights of the volume include: a history of Tennenbaum's theorem in arithmetic; a number of papers on Tennenbaum phenomena in weak arithmetics as well as on other aspects of arithmetics, such as interpretability; the transcript of Gödel's previously unpublished 1972–1975 conversations with Sue Toledo, along with an appreciation of the same by Curtis Franks; Hugh Woodin's paper arguing against the generic multiverse view; Anne Troelstra's history of intuitionism through 1991; and Aki Kanamori's history of the Suslin problem in set theory. The book provides a historical and philosophical treatment of particular theorems in arithmetic and set theory, and is ideal for researchers and graduate students in mathematical logic and philosophy of mathematics.
eBook
DISJUNCTIONS WITH STOPPING CONDITIONS
We introduce a tool for analysing models of CT⁻, the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan's theorem that the arithmetical part of models of CT⁻ are recursively saturated. We also use this tool to provide a new proof of theorem from [8] that all models of CT⁻ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for a set of formulae whose syntactic depth forms a nonstandard cut which cannot be extended to a full truth predicate satisfying CT⁻.
Journal Article
Logic without borders
In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy and computer science continue to be important and noticeable. The series publishes conference proceedings as well as monographs written by leading researchers in mathematical logic.
eBook
Is there Ethics in Mathematics?
This is a critical response to some arguments and general recommendations presented in a discussion paper Four Levels of Ethical Engagement [EiM Discussion Paper 1/2018 University of Cambridge Ethics in Mathematics Project, https://www.ethics-in-mathematics.com/assets/dp/18 1.pdf] by Maurice Chiodo and Piers Bursill-Hall. Much in their article is based on certain observations about characteristic psychological traits of mathematicians and their patterns of behavior that I find to be in stark contrast to my own observations. I argue against their assumptions and conclusions using examples.
Paper