相关论文: How to release Frege's system from Russell's antin…

Reverse Mathematics is a program in the foundations of mathematics which provides an elegant classification of theorems of ordinary mathematics based on computability. Our aim is to provide an alternative classification of theorems based on…

We revisit the mathematical foundations of proper scoring rules (PSRs) and Bregman divergences and present their characteristic properties in a unified theoretical framework. In many situations it is preferable not to generate a PSR…

Complex classifiers may exhibit "embarassing" failures in cases where humans can easily provide a justified classification. Avoiding such failures is obviously of key importance. In this work, we focus on one such setting, where a label is…

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very…

This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a…

Considerations based on the known relation between different characteristic classes for singular hypersufaces suggest that a form of the `inclusion-exclusion' principle may hold for Segre classes. We formulate and prove such a principle for…

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a…

Let No be Conway's class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some "tameness" and uniformity condition, f must satisfy some interesting…

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself),…

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the…

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…

We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value…

We present an elementary proof of the prime number theorem. The relative error follows a golden ratio scaling law and respects the bound obtained from the Riemann's hypothesis. The proof is derived in the framework of a scale free…

Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly, on even very simple programs,…

In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of…

G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…

The categorical models of the differential lambda-calculus are additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential…

Wilhelm (2021) has recently defended a criterion for comparing structure of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM * , another widely adopted criterion. We argue that this is mistaken;…

Classification is a ubiquitous and fundamental problem in artificial intelligence and machine learning, with extensive efforts dedicated to developing more powerful classifiers and larger datasets. However, the classification task is…

The statement of E.R. Loubenets, Phys. Rev. A 69, 042102 (2004), that separable states can violate classical probabilistic constraints is based on a misleading definition of classicality, which is much narrower than Bell's concept of local…