Related papers: Nonstandard proof methods in toposes
Modalities in homotopy type theory are used to create and access subuniverses of a given type universe. These have significant applications throughout mathematics and computer science, and in particular can be used to create universes in…
We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space $X=(\mathbb R^n ,\|\cdot\| )$ there exists an invertible linear map $T:\mathbb R^n \to \mathbb R^n$ with \[…
As the prototypical category, $\mathbf{Set}$ has many properties which make it special amongst categories. From the point of view of mathematical logic, one such property is that $\mathbf{Set}$ has enough structure to "properly" formalise…
In applied settings, tests of hypothesis where a nuisance parameter is only identifiable under the alternative often reduces into one of Testing One Hypothesis Multiple times (TOHM). Specifically, a fine discretization of the space of the…
Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system. We use the classical principle of extracting the algorithm as…
Let $(X,\rho,G)$ be a $G-$action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. We prove a variational principle for topological entropy of saturated sets for systems which have…
The connections among natural language processing and argumentation theory are becoming stronger in the latest years, with a growing amount of works going in this direction, in different scenarios and applying heterogeneous techniques. In…
A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then…
By using nonstandard analysis, we prove embeddability properties of difference sets $A-B$ of sets of integers. (A set $A$ is "embeddable" into $B$ if every finite configuration of $A$ has shifted copies in $B$.) As corollaries of our main…
In [44], we qualitatively studied some classical results implied by the specification property for dynamical systems with non-uniform specification. In this paper, we perform quantitative studies on how properties of topological theory and…
We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over $\mathrm{NP}$. Our main results concern $\mathrm{DP}$, i.e., the second level of this hierarchy: If all sets in $\mathrm{DP}$ have…
We give a model-theoretic characterisation of the geometric theories classified by \'etendues -- the `locally localic' topoi. They are the theories where each model is determined, syntactically and semantically, by any witness of a fixed…
Any scheme has its associated little and big Zariski toposes. These toposes support an internal mathematical language which closely resembles the usual formal language of mathematics, but is "local on the base scheme": For example, from the…
To each simplicial set $X$ we naturally assign an \'etendue ${\'E X}$ whose internal logic captures information about the geometry of $X$. In particular, we show that, for 'non-singular' objects $X$ and $Y$, the \'etendues ${\'E X}$ and…
A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including…
The idea of this approach towards proving the consistency of Quine's New Foundations set theory is to go in a completely untyped manner. So no contemplation about types is utilized here. All conceptualization pivots around proving a handful…
We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We…
We seek progress in the study of subtoposes of the effective topos. First we treat Van Oosten's result that local operators on the effective topos are internally NNO-indexed joins of what we shall call 'basic' local operators. Our main…
Consider a set represented by an inequality. An interesting phenomenon which occurs in various settings in mathematics is that the interior of this set is the subset where strict inequality holds, the boundary is the subset where equality…
Current astrophysical models of the interstellar medium assume that small scale variation and noise can be modelled as Gaussian random fields or simple transformations thereof, such as lognormal. We use topological methods to investigate…