Mathematical Logic Quarterly最新文献

We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.

Assume $�\mathrm{ZFC}$. Let κ be a cardinal. A -groundis a transitive proper class W modelling $�\mathrm{ZFC}$ such that V is a generic extension of W via a forcing $��P�\in �W�$ of cardinality . The κ-mantle $�M_{\kappa }$ is the intersection of all -grounds. We prove that certain partial choice principles in $�M_{\kappa }$ are the consequence of κ being inaccessible/weakly compact, and some other related facts.

We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom $�\mathrm{UA}$ (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.

‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid $��(�N�,�·�)�$. We give a full characterization of the $�⌀$-definable stably embedded sets of T, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries but not elimination of finite imaginaries.

We show under $��\mathrm{ZF}�+�\mathrm{DC}�+�{\mathrm{AD}}_R�$ that every set of reals is I-regular for any σ-ideal I on the Baire space $�{\omega }^{\omega }$ such that $�P_I$ is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under $��\mathrm{ZF}�+�\mathrm{DC}�+�{\mathrm{AD}}^{+}�$ if we additionally assume that the set of Borel codes for I-positive sets is $�{\underset{˜}{\Delta }}_1^2$. If we do not assume $�\mathrm{DC}$, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch [2], we show under $��\mathrm{ZF}�+�{\mathrm{DC}}_R�$ without using $�\mathrm{DC}$ that every set of reals is I-regular for any σ-ideal I on the Baire space $�{\omega }^{}$

We prove that in the Cohen extension adding ℵ₃ generic reals to a model of $��\mathrm{ZFC}�+�\mathrm{CH}�$ containing a simplified (ω₁, 2)-morass, gap-2 morass-definable η₁-orderings with cardinality ℵ₃ are order-isomorphic. Hence it is consistent that $��2^{{\aleph }_0}�=�{\aleph }_3�$ and that morass-definable η₁-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $�R$ over ω that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinite back-and-forth construction of order-type ω₁ to an order-preserving bijection between objects of cardinality ℵ₃.

We show that for any $��k�\in �\omega �$, the structure $��⟨�H_k�,�\in �⟩�$ of sets that are hereditarily of size at most k is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure $��V_{\omega }�=�{\bigcup }_k�H_k�$ of hereditarily finite sets, which is well known to be bi-interpretable with the standard model of arithmetic $��⟨�N�,�+�,�·�⟩�$.

Given sets $��X�,�Y�$ and a regular cardinal μ, let $��\Phi �(�X�,�Y�,�\mu �)�$ be the statement that for any function $��f�:�X�\times �Y�\to �\mu �$, there are functions $��g_1�:�X�\to �\mu �$ and $��g_2�:�Y�\to �\mu �$ such that for all $��(�x�,�y�)�\in �X�\times �Y�$, $��f��(�x�,�y�)��\le �\max �\{�g_1��\left(�x�\right)�$