We consider ω n -automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ω n for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω 2 -automatic (resp. ω n -automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not (...) determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ω n -automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a ${\mathrm{\Sigma }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ nor a ${\mathrm{\Pi }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ . We obtain that there exist infinitely many ω n -automatic, hence also ω-tree-automatic, atomless boolean algebras $\mathcal{B}_{n}$ , n ≥ 1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14]. (shrink)

A complete Boolean algebra ${\mathbb{B}}$ satisfies property ${(\hbar)}$ iff each sequence x in ${\mathbb{B}}$ has a subsequence y such that the equality lim sup z n = lim sup y n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of (...) property ${(\hbar)}$ with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}$ is not a theorem of ZFC and that there is no cardinal ${\mathfrak{k}}$ , definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}$ is a theorem of ZFC. Also, we show that the set ${\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}$ is equal to ${[0, \mathfrak{h})}$ or ${[0, {\mathfrak h}]}$ and that both values are consistent, which, with the known equality ${{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}$ completes the picture. (shrink)

Trees are natural generalizations of ordinals and this is especially apparent when one tries to find an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals.

We show that both Rado's Conjecture and strong Chang's Conjecture imply that there are no special ℵ2-Aronszajn trees if the Continuum Hypothesis fails. We give similar result for trees of higher heights and we also investigate the influence of Rado's Conjecture on square sequences.

We show that both Rado's Conjecture and strong Chang's Conjecture imply that there are no special ℵ2-Aronszajn trees if the Continuum Hypothesis fails. We give similar result for trees of higher heights and we also investigate the influence of Rado's Conjecture on square sequences.

This is a continuation of our paper where we show that Rado's Conjecture can trivialize ‐sequences in some cases when ϑ is not necessarily a successor cardinal.

We give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further (...) research. (shrink)

We examine the differences between three standard classes of forcing notions relative to the way they collapse the continuum. It turns out that proper and semi-proper posets behave differently in that respect from the class of posets that preserve stationary subsets of \.

In this paper we start the analysis of the class, the class of cofinal types of directed sets of cofinality at most ℵ2. We compare elements of using the notion of Tukey reducibility. We isolate some simple cofinal types in, and then proceed to find some of these types which have an immediate successor in the Tukey ordering of.

We use ideas of Fred Galvin to show that under Martin's axiom, there is a prime ideal on Pω (λ) with the partition property for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}.

Local sentences were introduced by Ressayre in [6] who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact strength was left open in [4]. Here we solve this problem, using a combinatorial result of J. H. Schmerl [7]. In fact, we show that the stretching (...) principles are equivalent to the existence of n -Mahlo cardinals for appropriate integers n. This is done by proving first that for every integer n, there is a local sentence φn having well ordered models of order type τ, for every infinite ordinal τ > ω which is not an n -Mahlo cardinal. (shrink)

Trees are natural generalizations of ordinals and this is especially apparent when one tries to find an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals.

In this paper we will analyze Baumgartner’s problem asking whether it is consistent that \ and every pair of \-dense subsets of \ are isomorphic as linear orders. The main result is the isolation of a combinatorial principle \\) which is immune to c.c.c. forcing and which in the presence of \ implies that two \-dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC (...) that there exists an \ dense suborder X of \ which cannot be embedded into \ in any outer model with the same \. (shrink)

Todorčević (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for (...) pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$. (shrink)

One of the main problems about is that whether a coherent Suslin tree forces that there are no S‐spaces under. We analyze a forcing notion related to this problem, and show that under, S forces that every topology on ω1 generated by a basis in the ground model is not an S‐topology. This supplements the previous work due to Stevo Todorčević [25].

v. 1. Philosophie. Rechtsphilosophie -- v. 2-4. Volkenrecht -- v. 5-6. Rechtsgeschiedenis. Burgerlijk recht. Procesrecht. Octrooirecht. Studie.

In lieu of an abstract, here is a brief excerpt of the content:Bonaventure's Reductio of the Nine Choirs of Angels:How Bonaventure Compressed Two Monumental Traditions into Nine Words and Nine Short PhrasesRandall B. Smith"There is probably no better illustration in medieval thought of how the genius of the symbolic imagination also involves deep speculative insight." So wrote Bernard McGinn of Bonaventure's Itinerarium mentis in deum in The Flowering of Mysticism: Men and Woman in the New Mysticism, 1200–1350.1 There is no (...) question that the Itinerarium is a book suffused with symbolic imagery; so much so, in fact, that it can sometimes seem to lapse into a confusing jumble.Certainly there are many sections of the Itinerarium that have baffled readers over the years, often because Bonaventure's use of imagery is so unique and creative and the truths he is attempting to enunciate are so far beyond our normal ways of thinking about God. But there is a short passage in chapter 4 of the Itinerarium that seems to defy interpretation precisely because of what one might describe as an almost austere terseness and brevity. The point Bonaventure wishes to make is one he communicates in little more than a list of nine words and nine short, parallel phrases.The passage I have in mind is this one in Itinerarium 4.4.When these things have been accomplished, and our spirit has been brought into conformity with the heavenly Jerusalem, it is ordered [End Page 583] hierarchically so that it can ascend upward. For no one enters into that city unless that city has first descended into the person's heart by means of grace, as John sees in his Apocalypse. It descends into the heart when our spirit has been made hierarchical by the reformation of the image, by means of the theological virtues, the enjoyment of the spiritual senses, and the ecstasy of rapture; for it has been purged, illumined and brought to perfection.In this way our spirit is adorned with nine orderly levels when within it the following are found in an appropriate order: announcing, dictating, leading, ordering, strengthening, commanding, receiving, revealing, and anointing. These correspond to the nine choirs of angels. The first three of the foregoing levels in the human mind relate to nature; the next three relate to work; and the final three relate to grace. When it has attained these, the soul, by entering into itself, enters into the heavenly Jerusalem where, as it considers the choirs of angels, it sees in them the God who dwells in them and who works in all their operations.Therefore, Bernard says to Pope Eugene that "God loves in the Seraphim as charity; knows in the Cherubim as truth; sits in the Thrones as justice; reigns in the Dominations as majesty; rules in the Principalities as a guiding principle; protects in the Powers as salvation; is at work in the Virtues as strength; reveals in the Archangels as light; assists in the Angels as kindness." From all this, God is seen as all in all when we contemplate God in our minds where God dwells through the gifts of the most generous charity.2Notice in the passage I have quoted above that Bonaventure says our spirit must be "brought into conformity with the heavenly Jerusalem," and that this requires it be "ordered hierarchically so that it can ascend upward." This serves as a transition into a rather odd pair of lists—one with nine words and the other with nine short, parallel phrases—that Bonaventure associates with the nine angelic choirs. These two lists of the nine choirs of angels do not, however, correspond exactly with one [End Page 584] another, for reasons I will explain below. And although these two lists of the nine ranks of angels in the angelic hierarchy take up just a few lines in Bonaventure's text, there is a lot of background information we will need to fill in to provide the necessary context to appreciate them. Indeed, as we will see, in this short passage, Bonaventure was able using the methods of the thirteenth-century sermo modernus style to reduce two monumental commentary traditions regarding the nine choirs of angels—one that... (shrink)

In this book, Yaqub describes a simple conception of truth and shows that it yields a semantical theory that accommodates the whole range of our seemingly conflicting intuitions about truth. This conception takes the Tarskian biconditionals as correctly and completely defining the notion of truth. The semantical theory, which is called the revision theory, that emerges from this conception paints a metaphysical picture of truth as a property whose applicability is given by a revision process rather than by a fixed (...) extension. The main advantage of this revision process is its ability to explain why truth seems in many cases almost redundant, in others substantial, and yet in others paradoxical. Yaub offers a comprehensive defense of the revision theory of truth by developing consistent and adequate formal semantics for languages in which all sorts of problematic sentences can be constructed. Yaqub concludes by introducing a logic of truth that further demonstrates the adequacy of the revision theory. (shrink)