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We construct a model in which the tree property holds in ${\aleph _{\omega + 1}}$ and it is destructible under $Col\left( {\omega ,{\omega _1}} \right)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.

I construct, in ZFC, a forcing notion that collapses $\aleph _3 $ and preserves all other cardinals. The existence of such a forcing answers a question of Uri Abraham from 1983.

We show that from large cardinals it is consistent to have the tree property simultaneously at${\aleph _{{\omega ^2} + 1}}$and${\aleph _{{\omega ^2} + 2}}$with${\aleph _{{\omega ^2}}}$strong limit.

Heretofore three fragments of Maniʼs Living Gospel – the most important work of Mani – have been recognized in Middle Persian and in Manichaean script: M17, M172/I/ and M644. This article, with a codicological and textological approach, shows that M17 and the new fragment M5439 are two separate pieces of a single manuscript page. The verso-side of M5439 has a new text. Now, after the identification and reading of this damaged fragment, we are able to correct the previous reconstructions and comment on a few Middle Persian words. The text contains a part of the exordium and of the chapter Aleph of the Living Gospel.

Measuring says that for every sequence ${\left( {{C_\delta }} \right)_{\delta < {\omega _1}}}$ with each ${C_\delta }$ being a closed subset of δ there is a club $C \subseteq {\omega _1}$ such that for every $\delta \in C$ , a tail of $C\mathop \cap \nolimits \delta$ is either contained in or disjoint from ${C_\delta }$ . We answer a question of Justin Moore by building a forcing extension satisfying measuring together with ${2^{{\aleph _0}}} > {\aleph _2}$ . The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable structures as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:{\omega _1} \to {\omega _1}$ mod. countable at each stage.

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More broadly, we define a notion of $\kappa$-concrete AEC---an AEC-like category in which only the $\kappa$-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [LR] yield a proof that any categorical mAEC is $\mu$-d-stable in many cardinals below the categoricity cardinal., Comment: v2: changed terminology. v3: tightened inequalities. v4: clarifying notes added. v5: referee's comments incorporated, with substantial improvements

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.

The cliché that a picture is worth a thousand words vastly underestimates the expressive power of many pictures and diagrams. In this note we show that even a simple map such as the outline of Manhattan Island, accompanied by a pointer marking North, implies a vast infinity of true statements.

The existence of a countable complete theory with exactly ℵ1 many minimal models is independent of ZFC + ¬CH.

If R is a complete discrete valuation ring and M is a reduced, torsion-free R -module of rank \kappa , where \aleph_0 \leq \kappa , we show that M \prop\oplus_(\aleph_0) R \oplus C for some R -module C . As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R) , where \alpha \leq \aleph_0 , and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0) , where Fin M denotes the set of endomorphisms of M with finite rank image.

Here an example will be given of a complete Lω1ω-sentence with a model of power ℵ1 but with no model of higher power. The continuum hypothesis is not assumed. The question of whether such an example exists was brought to the author's attention by Professor M. Makkai.An Lω1ω-sentence is said to be complete if its models all satisfy the same Lω1ω-sentences, or, equivalently, if all of the countable L-structures satisfying the sentence are isomorphic. Scott [5] showed that any countable L -structure (where L is countable) must satisfy a complete Lω1ω-sentence. Such a sentence is called a Scott sentence for the structure. An uncountable L-structure need not satisfy any complete Lω1ω -sentence.A complete Lω1ω-sentence σ is said to characterize the infinite cardinal k if σ has a model of power k but not of any higher power. The set of cardinals characterized by complete Lω1ω -sentences will be denoted by CC. By a result of Lopez-Escobar [3], if k ∈ CC, k ω1.Assuming GCH (so that ⊐α = ℵα, ), Malitz [4] showed that CC = {ℵα: α < ω1}.Without assuming GCH, Baumgartner [1] showed that ⊐α ∈ CC for all α < ω1.Without GCH, it is unknown whether ℵn ∈ CC for n ≥ 2. Now it will be shown that ℵ1, ∈ CC.

Let n ≥ 3. The following theorems are proved.Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable.Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that K ⊨ φ if and only if R φ σ(φ).Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ1-categorical.

In recent years, the Axiom of Determinateness (AD) has yielded numerous results concerning the size and properties of the first ω-many uncountable cardinals. Briefly, these results began with Solovay's discovery that ℵ1 and ℵ2 are measurable [8], [3], continued with theorems of Solovay, Martin, and Kunen concerning infinite-exponent partition relations [6], [3], Martin's proof that ℵn has confinality ℵ2 for 1 < n < ω, and very recently, Kleinberg's proof that the ℵn are Jonsson cardinals [4].This paper was inspired by a very recent result of Martin from AD that ℵ1 is ℵ2-super compact. It was known for some time that AD implies ℵ1 is α-strongly compact for all ℵ < θ (where θ is the least cardinal onto which 2ω cannot be mapped, quite a large cardinal under AD), and that ADR implies that ℵ1, is α-super compact for all α < θ. A key open question had been whether or not ℵ1 is super compact under AD alone.This paper comments on the method of Martin in several different ways. In §2, we will prove that ℵ1 is ℵ2-super compact, and then generalize the method to show that ℵ2 is ℵ3-strongly compact. In addition, we will demonstrate a limitation in the method by showing that the possible measures obtained on are not normal, and that the method cannot be extended to show that ℵ2 is ℵ4-strongly compact.

One problem in set theory without the axiom of choice is to find a reasonable way of estimating the size of a non-well-orderable set; in this paper we present evidence which suggests that this may be very hard. Given an arbitrary fixed aleph κ we construct a model of set theory which contains a set X such that if Y ⊆ X then either Y or X - Y is finite, but such that κ can be mapped into S(S(S(X))). So in one sense X is large and in another X is one of the smallest possible infinite sets. (Here S(X) is the power set of X.)

A quick glance through the commentaries and translations will show that most scholars see no serious textual problems in Isa 28:12. Laberge has suggested that “he said,” be corrected to “I kept saying,” and everyone notes the extra aleph at the end of “they were (not) willing …,” but otherwise the text appears to be universally regarded as sound. The present writer does not share that optimism, however. Perhaps the following stichometric arrangement of the verse will convey the difficulty he perceives