Mathematics

This note was written in Jan. 23, 2015 to answer a problem raised by G. Moser, who asked a constructive proof of a theorem by Ferreira-Zantema.

Let R be a ring equipped with a proper norm. We show that under suitable conditions on R , there is a natural basis under continuous linear injection for the set of Polish R -modules which are not countably generated. When R is a division ring, this basis can be taken to be a singleton.

We define a general class of dependent type theories, encompassing Martin-Löf's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions to be given in reasonable generality, rather than just for specific theories. Compared to other approaches, our definition stays closer to the direct or naive reading of syntax, yielding the traditional presentations of specific theories as closely as possible. Specifically, we give three main definitions: raw type theories, a minimal setup for discussing dependently typed derivability; acceptable type theories, including extra conditions ensuring well-behavedness; and well-presented type theories, generalising how in traditional presentations, the well-behavedness of a type theory is established step by step as the type theory is built up. Following these, we show that various fundamental fitness-for-purpose metatheorems hold in this generality. Much of the present work has been formalised in the proof assistant Coq.

We prove a limitation on a variant of the KPT theorem proposed for propositional proof systems by Pich and Santhanam (2020), for all proof systems that prove the disjointness of two NP sets that are hard to distinguish.

We present a new manifestation of Gödel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension of Peano arithmetic ( PA ) by a mathematically meaningful axiom scheme that consists of Σ 0 2 -sentences. These sentences assert that each computably enumerable ( Σ 0 1 -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is important for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel's second incompleteness theorem. In contrast, Gödel's theorem is not needed for typical examples of Π 0 2 -independence (such as the Paris-Harrington principle), since computational strength provides an extensional invariant on the level of Π 0 2 -sentences.

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ♢ -based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ♢ . In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved.

The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by {\textbf{IL}}({\rm All}) . In this paper we present a new principle \sf R in {\textbf{IL}}({\rm All}) . We show that \sf R does not follow from the logic {\textbf{IL}}{\sf P_0W^*} that contains all previously known principles. This is done by providing a modal incompleteness proof of {\textbf{IL}}{\sf P_0W^*} : showing that \sf R follows semantically but not syntactically from {\textbf{IL}}{\sf P_0W^*} . Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle \sf R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories.

Suppose that Ω is a complex lattice that is closed under complex conjugation and that I is a small real interval, and that D is a disc in C . Then the restriction ℘ | D is definable in the structure ( R ¯ ,℘ | I ) if and only if the lattice Ω has complex multiplication. This characterises lattices with complex multiplication in terms of definability.

In this note we provide some applications of Löwenheim-Skolem cardinals introduced in \cite{U}.

A version of Woodin's HOD dichotomy is proved assuming the existence of just one strongly compact cardinal.