Archive for Mathematical Logic最新文献

We study the theory of K-vector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-fold sums of X with itself. If K is finite this is no longer true, but we still have that a natural completion is near-model-complete.

Let ({mathcal {R}}) be a polynomially bounded o-minimal expansion of the real field. Let f(z) be a transcendental entire function of finite order (rho ) and type (sigma in [0,infty ]). The main purpose of this paper is to show that if ((rho <1)) or ((rho =1) and (sigma =0)), the restriction of f(z) to the real axis is not definable in ({mathcal {R}}). Furthermore, we give a generalization of this result for any (rho in [0,infty )).

The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.

Suppose that (kappa ) is indestructibly supercompact and there is a measurable cardinal (lambda > kappa ). It then follows that (A_0 = {delta < kappa mid delta ) is a measurable cardinal and the Mitchell ordering of normal measures over (delta ) is nonlinear(}) is unbounded in (kappa ). If the Mitchell ordering of normal measures over (lambda ) is also linear, then by reflection (and without any use of indestructibility), (A_1= {delta < kappa mid delta ) is a measurable cardinal and the Mitchell ordering of normal measures over (delta ) is linear(}) is unbounded in (kappa ) as well. The large cardinal hypothesis on (lambda ) is necessary. We demonstrate this by constructing via forcing two models in which (kappa ) is supercompact and (kappa ) exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that (A_0) is unbounded in (kappa ) if (lambda > kappa ) is measurable. In one of these models, for every measurable cardinal (delta ), the Mitchell ordering of normal measures over (delta ) is linear. In the other of these models, for every measurable cardinal (delta ), the Mitchell ordering of normal measures over (delta ) is nonlinear.

When the Canonical Ramsey’s Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the (lambda )-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-(omega ) exponentiation is reducible to this same principle by a uniform computable reduction.

We present a cut-free sequent calculus for a class of first-order theories in negation normal form which include coherent and co-coherent theories alike. All structural rules, including cut, are admissible.

Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent L-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an L-domain, and every L-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of L-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting.

Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes (textrm{E}_k) and (textrm{U}_k) introduced in [1] are exactly the classes induced by (Sigma _k) and (Pi _k) respectively via the transformation procedure in any first-order theory.

We show that we can interpret concatenation theories in arithmetical theories without coding sequences by identifying binary strings with (2times 2) matrices with determinant 1.

In this paper, we study some new examples of ideals on (omega ) with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin–Keisler order.