We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ : For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)-compactness property and the Löwenheim-Skolem theorem down to κ . With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the Löwenhe...

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ: For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the LöwenheimS...

We find many conditions equivalent to the modeltheoretical property λ κ ⇒ μ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices. See Part I [L7] or [CN, CK, KM, KV, HNV] for unexplained notation. According to [L1], if λ ≥ μ are infinite regular cardinals, and κ is a cardina...

We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, for some regular cardinal ν. More precisely, we s...

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. Theorem 0.1 (Superstability from categoricity). Let K be a (< κ)-tame AEC with amalgamation. If κ = iκ > ...

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...

v. dannon showed that spherical curves in e4 can be given by frenet-like equations, and he thengave an integral characterization for spherical curves in e4 . in this paper, lorentzian spherical timelike andspacelike curves in the space time 41 r are shown to be given by frenet-like equations of timelike andspacelike curves in the euclidean space e3 and the minkowski 3-space 31 r . thus, finding...

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ-strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to cons...

Assume that κ and λ are respectively strong weakly compact cardinals with λ>κ. Fix Θ≥λ a cardinal cof(Θ)>κ cof(δ)=δ<κ. Assuming the GCH≥κ holds, we construct generic extension of universe where is limit cardinal, cof(κ)=δ, 2κ=Θ TP(κ++) holds. This extends main result [5] for uncountable cofinalities.