Archive for Mathematical Logic最新文献

A complete theory T has the Schröder–Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if T is a strictly stable theory then T does not have the SB-property.

This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to (mathrm ACA_{0}) over the base system (mathrm RCA_{0}). The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in (mathrm RCA_{0}); as an application, we formalize an effective version of the Yoneda Embedding in (mathrm RCA_{0}). Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for relational structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for all (n in mathbb {N} ), every subflow of the space (S_n(M)) of n-types has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.

In Bagaria (J Symb Log 81(2), 584–604, 2016), Bagaria and Väänänen developed a framework for studying the large cardinal strength of downwards Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis, originally introduced by the third author in Väänänen (Applications of set theory to generalized quantifiers. PhD thesis, University of Manchester, 1967); Väänänen (in Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 391–421. North-Holland, Amsterdam 1979) Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of symbiosis and apply it to upwards Löwenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of symbiosis to a new form, called bounded symbiosis. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards Löwenheim–Skolem-type principles for second order logic.

We explore approximate categoricity in the context of distortion systems, introduced in our previous paper (Hanson in Math Logic Q 69(4):482–507, 2023), which are a mild generalization of perturbation systems, introduced by Yaacov (J Math Logic 08(02):225–249, 2008). We extend Ben Yaacov’s Ryll-Nardzewski style characterization of separably approximately categorical theories from the context of perturbation systems to that of distortion systems. We also make progress towards an analog of Morley’s theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal (kappa ) such that every model of size (kappa ) is ‘approximately saturated,’ in the appropriate sense, then the same is true for all uncountable cardinalities. Finally we present some examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.

Using the Feferman-Vaught Theorem, we prove that a definable subset of a product structure must be a Boolean combination of open sets, in the product topology induced by giving each factor structure the discrete topology. We prove that for families of structures with certain properties, including families of integral domains, the pure Boolean generalized product is definable in the direct product structure. We use these results to obtain characterizations of the definable subsets of (prod _p mathbb {F}_p)—in particular, every formula is equivalent to a Boolean combination of (exists forall exists ) formulae.

In the analysis of the blurry (textsf{HOD}) hierarchy, one of the fundamental concepts is that of a leap, and it turned out that critical leaps are of particular interest. A critical leap is a leap which is the cardinal successor of a singular strong limit cardinal. Such a leap is sudden if its cardinal predecessor is not a leap, and otherwise, it is smooth. In prior work, I showed that the existence of a sudden critical leap is equiconsistent with the existence of a measurable cardinal. Here, I show that if the cofinality of the cardinal predecessor of a sudden critical leap is required to be uncountable, the consistency strength increases considerably. I also show that when focusing on critical leaps whose cardinal predecessors have uncountable cofinality, the consistency strength of a smooth critical leap is much lower than that of a sudden critical leap. Finally, I observe that in contrast to the countable cofinality setting, (aleph _{omega _1+1}), e.g., cannot be a sudden critical leap.

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type (Sigma ). We introduce the notion of a Lascar tuple for (Sigma ) and by considering the space of types over a Lascar tuple for (Sigma ), the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type (Sigma ). Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types.

We investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)–(iii) are equivalent:

1. (i)

   (cf({omega }_1)={omega }_1),

2. (ii)

   ({omega }_1rightarrow ({omega }_1,{omega }+1)^2),

3. (iii)

   any family (mathcal {A}subset [{On}]^{<{omega }}) of size ({omega }_1) contains a (Delta )-system of size ({omega }_1).

Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as

1. (1)

   ({{omega }_2}rightarrow ({omega }_1,{omega }+1)),

2. (2)

   any family (mathcal {A}subset [{On}]^{<{omega }}) of size ({omega }_2) contains a (Delta )-system of size ({omega }_1).

Some statements can be proven in ZF using purely combinatorial arguments, such as:

1. (3)

   given a set mapping (F:{omega }_1rightarrow {[{omega }_1]}^{<{omega }}), the set ({omega }_1) has a partition into ({omega })-many F-free sets.

Other statements can be proven in ZF by employing certain methods of absoluteness, for example:

1. (4)

   given a set mapping (F:{omega }_1rightarrow {[{omega }_1]}^{<{omega }}), there is an F-free set of size ({omega }_1),

2. (5)

   for each (nin {omega }), every family (mathcal {A}subset {[{omega }_1]}^{{omega }}) with (|Acap B|le n) for ({A,B}in {[mathcal {A}]}^{2}) has property B.

In contrast to statement (5), we show that the following ZFC theorem of Komjáth is not provabl