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Register a new userFrom geometry to geology: An invitation to mathematical pluralism through the phenomenon of independence
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Jonas Reitz
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This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets $V$, and it is in this universe that mathematics takes place. From this view, the purpose of set theory is learning the truth about $V$. It has become apparent, however, that the phenomenon of independence - those questions left unresolved by the axioms - holds a central place in the investigation. This paper introduces the notion of independence, explores the primary tool (soundness) for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics - a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice local neighborhoods of the multiverse that are amenable to first-order analysis, and emph{set-theoretic geology} studies just such a neighborhood, the collection of grounds of a given universe $V$ of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments.
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The goal of this article is to invite the reader to get to know and to get involved into higher Teichmuller theory by describing some of its many facets.
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to Konigs lemma, such as restrictions of Konigs lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak Konigs lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak Konigs lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak Konigs lemma and algorithmic randomness by showing that Ramsey-type weak weak Konigs lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak Konigs lemma. This answers a question of Flood.
This is a set of 288 questions written for a Moore-style course in Mathematical Logic. I have used these (or some variation) four times in a beginning graduate course. Topics covered are: propositional logic axioms of ZFC wellorderings and equivalents of AC ordinal and cardinal arithmetic first order logic, and the compactness theorem Lowenheim-Skolem theorems Turing machines, Churchs Thesis completeness theorem and first incompleteness theorem undecidable theories second incompleteness theorem
An electromagnetic theory of thermal radiation is outlined, based on the fluctuation electrodynamics of Rytov and co-workers. We discuss the basic concepts and the status of different approximations. The physical content is illustrated with a few examples on near-field heat transfer.
The variety generated by the Brandt semigroup ${bf B}_2$ can be defined within the variety generated by the semigroup ${bf A}_2$ by the single identity $x^2y^2approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${bf B}_2^1$ and ${bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${bf A}_2^1$ that satisfy $x^2y^2approx y^2x^2$ and contain ${bf B}_2^1$. A further consequence is that the variety of ${bf B}_2^1$ cannot be defined within the variety of ${bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2yapprox yx^2$ and containing the monoid $M({bf z}_infty)$, where ${bf z}_infty$ denotes the infinite limit of the Zimin words ${bf z}_0=x_0$, ${bf z}_{n+1}={bf z}_n x_{n+1}{bf z}_n$.
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