Suppose that \(\kappa \) is indestructibly supercompact and there is a measurable cardinal \(\lambda > \kappa \). It then follows that \(A_0 = \{\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 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. (shrink)

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.

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 $$\lambda $$ λ -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-$$\omega $$ ω exponentiation is reducible to this same principle by a uniform computable reduction. (shrink)

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.

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

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 ) 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.

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. (shrink)

We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic \(\textbf{IL}\). We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing (...) degrees. In particular, we show that every \(\Delta ^0_2\) -degree contains an NCR element. (shrink)