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Register a new userA symmetric $beta$-model
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Stephen G. Simpson
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We prove that there exists a countable $beta$-model in which, for all reals $X$ and $Y$, $X$ is definable from $Y$ if and only $X$ is hyperarithmetical in $Y$. We also obtain some related results and pose some related questions.
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Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras $bf A$. We provide a complete classification for the case that $bf A$ is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation $mathfrak{B}$. We can then study the computational complexity of the network satisfaction problem of ${bf A}$ using the universal-algebraic approach, via an analysis of the polymorphisms of $mathfrak{B}$. We also use a Ramsey-type result of Nev{s}etv{r}il and Rodl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.
We will prove that there exists a model of ZFC+``c= omega_2 in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq R of cardinality c there exists a continuous function f:R-> R with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R .
We construct a model of $mathsf{ZF} + mathsf{DC}$ containing a Luzin set, a Sierpi{n}ski set, as well as a Burstin basis but in which there is no a well ordering of the continuum.
We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consist of so-called emph{Mammen spaces}, where a Mammen space is a triple $(U,mathcal S,mathcal C)$, where $U$ is a non-empty set (the universe), $mathcal S$ is a perfect Hausdorff topology on $U$, and $mathcal Csubseteqmathcal P(U)$ together with $mathcal S$ satisfy certain axioms. We refute a conjecture put forward by J. Hoffmann-J{o}rgensen, who conjectured that the existence of a complete Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology. On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Finally, we investigate two new cardinal invariants $mathfrak u_M$ and $mathfrak u_T$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. We show $mathfrak u_M=mathfrak u_T=2^{aleph_0}$ follows from Martins Axiom, and, contrastingly, we show that $aleph_1=mathfrak u_M=mathfrak u_T<2^{aleph_0}=aleph_2$ in the Baumgartner-Laver model.
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