We consider a version of -Miller forcing on an uncountable cardinal . We show that under this forcing collapses to and adds a -Cohen real. The same holds under the weaker assumptions that , , and forcing with collapses to .

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility . This gives rise to a rich degree structure. In this paper, we lift the study of -degrees to the case. In doing so, we rely on the Ershov hierarchy. For any notation for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by on the equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of -degrees.

We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7-8):1045-1103, 2017. 10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń's diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals.

Lindström's Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, , as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

A Kaufmann model is an -like, recursively saturated, rather classless model of (or ). Such models were constructed by Kaufmann under the combinatorial principle and Shelah showed they exist in by an absoluteness argument. Kaufmann models are an important witness to the incompactness of similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be "killed" by forcing without collapsing . We show that the answer to this question is independent of and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of whether or not Kaufmann models can be axiomatized in the logic where is the quantifier "there exists uncountably many".

We study -maximal cofinitary groups for regular uncountable, . Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: Any -maximal cofinitary group has many orbits under the natural group action of on .If then any partition of into less than many sets can be realized as the orbits of a -maximal cofinitary group.For any regular it is consistent that there is a -maximal cofinitary group which is universal for groups of size . If we only require the group to be universal for groups of size then this follows from .

We show that Hechler's forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are -Canjar for any countably directed unbounded family  of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that in every extension by the above forcing notions.

We prove that if is an infinite Boolean algebra in the ground model and is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any -generic extension [], has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

For a free filter on , endow the space , where , with the topology in which every element of is isolated whereas all open neighborhoods of are of the form for . Spaces of the form constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson-Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter , the space carries a sequence of normalized finitely supported signed measures such that for every bounded continuous real-valued function on if and only if , that is, the dual ideal is Katětov below the asymptotic density ideal . Consequently, we get that if , then: (1) if is a Tychonoff space and is homeomorphic to a subspace of , then the space of bounded continuous real-valued functions on contains a complemented copy of the space endowed with the pointwise topology, (2) if is a compact Hausdorff space and is homeomorphic to a subspace of , then the Banach space () of continuous real-valued functions on is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space contains a non-trivial convergent sequence, then the space () is not Grothendieck.

We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent (AFSP) in his work on the -conjecture. A recent related property is the (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish: holds for an ascending sequence then there is an inner model with measurables of arbitrarily large Mitchell order below , that is: . A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales.