Journal of Mathematical Logic

In [Countable thin ${\mathrm{\Pi }}_1^0$ 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 ${\mathrm{\Pi }}_1^0$ classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin ${\mathrm{\Pi }}_1^0$ classes. We will prove in this paper that the c.e. degrees containing no members of thin ${\mathrm{\Pi }}_1^0$ classes are dense in the c.e. degrees. We will also prove that the c.e. degrees containing members of thin ${\mathrm{\Pi }}_1^0$ 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).

We prove that for any computable successor ordinal of the form $\alpha =\delta +2k$ $(\delta$ limit and $k\in \omega )$ there exists computable torsion-free abelian group $($TFAG$)$ that is relatively ${\mathrm{\Delta }}_{\alpha }^0$ -categorical and not ${\mathrm{\Delta }}_{\alpha -1}^0$ -categorical. Equivalently, for any such $\alpha$ there exists a computable TFAG whose initial segments are uniformly described by ${\mathrm{\Sigma }}_{\alpha }^c$ infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly ${\mathrm{\Sigma }}_{\alpha }^c$-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) ${\mathrm{\Delta }}_{\alpha }^0$-categorical TFAGs for arbitrarily large $\alpha$ 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 $0^{″}$-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.

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.

The notion of a $\delta$-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis (CH) that for each $\delta <{\omega }_2$, any $\delta$-generic sequence of P-points can be extended to an ${\omega }_2$-generic sequence. This shows that the CH implies that there is a chain of P-points of length ${𝔠}^{+}$ with respect to both Rudin–Keisler and Tukey reducibility. These results answer an old question of Andreas Blass.

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 $2^{\kappa }$-supercompact cardinal $\kappa$ 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.