Forcing extensions yield models of ZFC in which a long sequence of club subsets of ω 1 has the following property: every subsequence of size ℵ 1 has a finite intersection.

Louveau, A., S. Shelah and B. Velikovi, Borel partitions of infinite subtrees of a perfect tree, Annals of Pure and Applied Logic 63 271–281. We define a notion of type of a perfect tree and show that, for any given type τ, if the set of all subtrees of a given perfect tree T which have type τ is partitioned into two Borel classes then there is a perfect subtree S of T such that all subtrees of S of (...) type τ belong to the same class. This result simultaneously generalizes the partition theorems of Galvin-Prikry and Galvin-Blass. The key ingredient of the proof is the theorem of Halpern-Laüchli on partitions of products of perfect trees. (shrink)

We prove that compactness is equivalent to the amalgamation property, provided the occurrence number of the logic is smaller than the first uncountable measurable cardinal. We also relate compactness to the existence of certain regular ultrafilters related to the logic and develop a general theory of compactness and its consequences. We also prove some combinatorial results of independent interest.

A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah . Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S on an infinite cardinal μ are all of the form Sκ = the subgroup consisting of elements whose support has cardinality 20, cƒ 20 < κ, 0 < κ < 20, and κ = 0, we make a further analysis of the first (...) order theory of Sλ/Sκ. introducing many-sorted second order structures , all of whose sorts have cardinality at most 20, and in terms of which we can completely characterize the elementary theory of the groups Sλ/Sκ. (shrink)

ḥeleḳ 1. Osher ṿe-śimḥah -- ḥeleḳ 2. Emunah u-viṭaḥon -- ḥeleḳ 3. Ṿa-ani tefilati -- ḥeleḳ 4. Shuvah Yiśraʼel -- ḥeleḳ 5. Ṿe-ahavta le-reʻakha -- ḥeleḳ 6. Kibud horim -- ḥeleḳ 7. Kibud ḥakhamim -- ḥeleḳ 8. Ṭov u-meṭiv --.

A class K of structures is controlled if for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence classes . We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled 69–88).

After small forcing, any $ -closed forcing will destroy the supercompactness and even the strong compactness of κ.

We give an example of a countable theory $T$ such that for every cardinal $\lambda \geq \aleph_2$ there is a fully indiscernible set $A$ of power $\lambda$ such that the principal types are dense over $A$, yet there is no atomic model of $T$ over $A$. In particular, $T$ is a theory of size $\lambda$ where the principal types are dense, yet $T$ has no atomic model.

We show that it is consistent that the reaping number r is less than u , the size of the smallest base for an ultrafilter. To show that our forcing preserves certain ultrafilters, we prove a general partition theorem involving Ramsey ideals.

It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from [3]. If (...) A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)

Let be a class of models with a notion of ‘strong’ submodel and of canonically prime model over an increasing chain. We show under appropriate set-theoretic hypotheses that if K is not smooth , then K has many models in certain cardinalities. On the other hand, if K is smooth, we show that in reasonable cardinalities K has a unique homogeneous-universal model. In this situation we introduce the notion of type and prove the equivalence of saturated with homogeneous-universal.

[1] Yotse la-or ʻal yede ṭalmidaṿ shomʻe liḳḥo be-yerakh ha-eṭanim di-shenaṭ 2016 -- [2] Yotse la-or ʻal yede ṭalmidaṿ shomʻe liḳḥo be-yerakh ha-eṭanim di-shenaṭ 2018.

A class K of structures is controlled if, for all cardinals λ, the relation of L ∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the ω-independence property is not controlled.

Removing a black hole conic singularity by means of Kruskal representation is equivalent to imposing extensibility on the Kasner–Fronsdal local isometric embedding of the corresponding black hole geometry. Allowing for globally non-trivial embeddings, living in Kaluza–Klein-like M 5 × S 1 (rather than in standard Minkowski M 6 ) and parametrized by some wave number k, extensibility can be achieved for apparently “forbidden” frequencies ω in the range ω 1 (k) ≤ ω ≤ ω 2 (k). As k → 0, (...) ω 1, 2 (0) → ωH (e.g., ωH = 1/4M in the Schwarzschild case) such that the Hawking–Gibbons limit is fully recovered. The various Kruskal sheets are then viewed as slices of the Kaluza–Klein background. Euclidean k discreteness, dictated by imaginary time periodicity, is correlated with flux quantization of the underlying embedding gauge field. (shrink)

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion Q such that, in any Q-generic extension of the universe, there are non-isomorphic models M 1 and M 2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if `c.c.c.' is replaced by other cardinal-preserving adjectives. We also give (...) an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings. (shrink)

This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.

Halakha is the all-encompassing source of normative Jewish conduct, and values and ethics are among Judaism's most important and fundamental aims. In this volume, Rabbi Aharon Lichtenstein masterfully plots the interplay of law and ethics. Drawing on his vast knowledge of classic Jewish sources, the analytic depth of the Brisker tradition, and his wide grasp of Western scholarship and thought, Rabbi Lichtenstein charts how values are intertwined with the details of Halakha and inform its general worldview. Exploring such topics (...) as the relationship between individual and community, the extent to which Halakha encourages or demands altruism, and the role of humanistic concerns in practical halakhic determination, Rabbi Lichtenstein analyzes rulings of halakhic giants in context of the relevant source material. The result is a first-of-its-kind study that offers a lucid yet extensive presentation of detailed halakhic points that combine to form a broad tapestry of the most basic questions of values in Halakha. -- cover page 4. (shrink)

ABSTRACT The paper is a response to the present crisis of higher education as reflected in the fragmentation of the university, and its increasingly pevformative character. It is based on a criticism of Ronald Barnett's recent attempt to tackle this problem. While agreeing with Barnett's fundamentally radical approach to higher education, the paper criticises Barnetts view on three levels: the methodological, theoretical and practical. It ends with guidelines for an alternative radical approach which avoids the practical problems besetting Barnetts proposal.