Authors:Borisa Kuzeljevic, Dilip Raghavan Abstract: Journal of Mathematical Logic, Ahead of Print. The notion of a [math]-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis (CH) that for each [math], any [math]-generic sequence of P-points can be extended to an [math]-generic sequence. This shows that the CH implies that there is a chain of P-points of length [math] with respect to both Rudin–Keisler and Tukey reducibility. These results answer an old question of Andreas Blass. Citation: Journal of Mathematical Logic PubDate: 2018-03-19T09:06:55Z DOI: 10.1142/S0219061318500046

Authors:Gabriel Goldberg Abstract: Journal of Mathematical Logic, Ahead of Print. We show from an abstract comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on sufficiently strong ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. This gives a conditional answer to the well-known question of whether a [math]-supercompact cardinal [math] must carry more than one normal measure of order 0. Conditioned on a very plausible iteration hypothesis, the answer is no, since the Ultrapower Axiom holds in the canonical inner models at the finite levels of supercompactness. Citation: Journal of Mathematical Logic PubDate: 2018-03-19T09:06:54Z DOI: 10.1142/S0219061318500058

Authors:Giorgio Audrito, Matteo Viale Abstract: Journal of Mathematical Logic, Volume 17, Issue 02, December 2017. The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [math] for a class of forcings [math] and a given ordinal [math]), and show that [math] implies generic absoluteness for the first-order theory of [math] with respect to forcings in [math] preserving the axiom, where [math] is a cardinal which depends on [math] ([math] if [math] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [math] with respect to [math] and for [math] with respect to [math] with [math] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [math] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author. Citation: Journal of Mathematical Logic PubDate: 2017-11-28T06:44:47Z DOI: 10.1142/S0219061317500052

Authors:Omer Ben-Neria, Spencer Unger Abstract: Journal of Mathematical Logic, Volume 17, Issue 02, December 2017. We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [math] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [math] in [math] is [math]-supercompact in HOD. Citation: Journal of Mathematical Logic PubDate: 2017-11-28T06:44:44Z DOI: 10.1142/S0219061317500076

Authors:Yair Hayut, Chris Lambie-Hanson Abstract: Journal of Mathematical Logic, Volume 17, Issue 02, December 2017. We investigate the relationship between weak square principles and simultaneous reflection of stationary sets. Citation: Journal of Mathematical Logic PubDate: 2017-11-28T06:44:43Z DOI: 10.1142/S0219061317500106

Authors:Rodney G. Downey, Alexander G. Melnikov, Keng Meng Ng Abstract: Journal of Mathematical Logic, Volume 17, Issue 02, December 2017. We solve a problem posed by Goncharov and Knight (Problem 4 in [S. Goncharov and J. Knight, Computable structure and antistructure theorems, Algebra Logika 41(6) (2002) 639–681, 757]). More specifically, we produce an effective Friedberg (i.e. injective) enumeration of computable equivalence structures, up to isomorphism. We also prove that there exists an effective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures. Citation: Journal of Mathematical Logic PubDate: 2017-11-28T06:44:42Z DOI: 10.1142/S0219061317500088

Authors:Toshimichi Usuba Abstract: Journal of Mathematical Logic, Volume 17, Issue 02, December 2017. A transitive model [math] of ZFC is called a ground if the universe [math] is a set forcing extension of [math]. We show that the grounds of[math][math][math] are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) [math] has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground. Citation: Journal of Mathematical Logic PubDate: 2017-11-28T06:44:40Z DOI: 10.1142/S021906131750009X

Authors:Antonio Montalbán Abstract: Journal of Mathematical Logic, Volume 17, Issue 02, December 2017. We prove Fraïssé’s conjecture within the system of [math]-comprehension. Furthermore, we prove that Fraïssé’s conjecture follows from the [math]-bqo-ness of 3 over the system of Arithmetic Transfinite Recursion, and that the [math]-bqo-ness of 3 is a [math]-statement strictly weaker than [math]-comprehension. Citation: Journal of Mathematical Logic PubDate: 2017-11-28T06:44:39Z DOI: 10.1142/S0219061317500064

Authors:Alexander G. Melnikov Abstract: Journal of Mathematical Logic, Ahead of Print. We prove that for any computable successor ordinal of the form[math][math][math][math] limit and[math][math] there exists computable torsion-free abelian group[math][math]TFAG[math] that is relatively[math][math]-categorical and not[math][math]-categorical. Equivalently, for any such [math] there exists a computable TFAG whose initial segments are uniformly described by [math] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [math]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively)[math][math]-categorical TFAGs for arbitrarily large [math] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [math]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions. Citation: Journal of Mathematical Logic PubDate: 2017-11-24T03:05:15Z DOI: 10.1142/S0219061318500022

Authors:Daniel M. Hoffmann, Piotr Kowalski Abstract: Journal of Mathematical Logic, Ahead of Print. We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action. Citation: Journal of Mathematical Logic PubDate: 2017-11-24T03:05:15Z DOI: 10.1142/S0219061318500034

Authors:Rodney G. Downey, Guohua Wu, Yue Yang Abstract: Journal of Mathematical Logic, Ahead of Print. In [Countable thin [math] classes, Ann. Pure Appl. Logic 59 (1993) 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees (not necessarily c.e.) containing members of countable thin [math] classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin [math] classes. We will prove in this paper that the c.e. degrees containing no members of thin [math] classes are dense in the c.e. degrees. We will also prove that the c.e. degrees containing members of thin [math] classes are dense in the c.e. degrees, improving the result of Cenzer et al. mentioned above. Thus, we obtain a new natural subclass of c.e. degrees which are both dense and co-dense in the c.e. degrees, while the other such class is the class of branching c.e. degrees (See [P. Fejer, The density of the nonbranching degrees, Ann. Pure Appl. Logic 24 (1983) 113–130] for nonbranching degrees and [T. A. Slaman, The density of infima in the recursively enumerable degrees, Ann. Pure Appl. Logic 52 (1991) 155–179] for branching degrees). Citation: Journal of Mathematical Logic PubDate: 2017-11-14T06:37:37Z DOI: 10.1142/S0219061318500010