Related papers: Vop\v{e}nka's Alternative Set Theory as a framewor…
Vop\v{e}nka's Alternative Set Theory can be viewed both as an evolution and as a revolution: it is based on his previous experience with nonstandard universes, inspired by Skolem's construction of a nonstandard model of arithmetic, and its…
It is shown that Vop\v{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product,…
Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set…
An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept "accessibility" is used coherently within finite set theory whose…
Variational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often…
We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings,…
We point out that the recursive formula that appears in Erickson's presentation "Fusible Numbers" is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we…
In this article, the interplay between Vop\v{e}nka's principle, as well as its weaker counterpart, and presentable $\infty$-categories is studied. Analogous statements, arising after replacing categories with $\infty$-categories in the…
Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments…
To quantify uncertainties in inverse problems of partial differential equations (PDEs), we formulate them into statistical inference problems using Bayes' formula. Recently, well-justified infinite-dimensional Bayesian analysis methods have…
We study system design problems stated as parameterized stochastic programs with a chance-constraint set. We adopt a Bayesian approach that requires the computation of a posterior predictive integral which is usually intractable. In…
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…
A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…
This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative…
A recent proposal for a superdeterministic account of quantum mechanics, named Invariant-set theory, appears to bring ideas from several diverse fields like chaos theory, number theory and dynamical systems to quantum foundations. However,…
We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…
Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal…
This paper is a contribution to the study of extensions of arbitrary models of ZF (Zermelo-Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. We present some new constructions of certain types…
Data vectors generalise finite multisets: they are finitely supported functions into a commutative monoid. We study the question if a given data vector can be expressed as a finite sum of others, only assuming that 1) the domain is…
This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order…