Archive for Mathematical Logic最新文献

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