Related papers: Rethinking set theory
Intuitive Set Theory (IST) is defined as the theory we get, when we add Axiom of Monotonicity and Axiom of Fusion to Zermelo-Fraenkel set theory. In IST, Continuum Hypothesis is a theorem, Axiom of Choice is a theorem, Skolem paradox does…
The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of…
A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative…
Humans are extremely swift learners. We are able to grasp highly abstract notions, whether they come from art perception or pure mathematics. Current machine learning techniques demonstrate astonishing results in extracting patterns in…
In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets.
The main goal of "Naive Axiomatic Mengenlehre" (NAM) is to find a more or less adequately explicit criterion that precisely formalizes the intuitive notion of a "normal set". NAM is mainly a construction procedure for building several…
The new axiom in set theory, axiom of the empty sets, allows another way to formulate Bell inequality in quantum mechanics. The new axiom emerges from the historical and philosophical analysis of set theory.
Simple type theory is formulated for use with the generic theorem prover Isabelle. This requires explicit type inference rules. There are function, product, and subset types, which may be empty. Descriptions (the eta-operator) introduce the…
As autonomous systems are increasingly deployed in open and uncertain settings, there is a growing need for trustworthy world models that can reliably predict future high-dimensional observations. The learned latent representations in world…
Understanding the notion of a model is not always easy in logic courses. Hence, tools such as Euler diagrams are frequently applied as informal illustrations of set-theoretical models. We formally investigate Euler diagrams as an…
ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sentences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this…
Cognitive theories for reasoning are about understanding how humans come to conclusions from a set of premises. Starting from hypothetical thoughts, we are interested which are the implications behind basic everyday language and how do we…
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…
It is well known that ZFC, despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the…
In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of…
We look at various features of the Standard Model with the purpose of exploring some possibilities of how to seek physical laws beyond it, i.e. at even smaller distances. Only parameters and structure which are not calculable from the…
The sets used to construct other mathematical objects are pure sets, which means that all of their elements are sets, which are themselves pure. One set may therefore be within another, not as an element, but as an element of an element, or…
We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a…
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…
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a…