Related papers: Labeled Mereological Set Theory
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
The paper is a naive introduction to descriptive set theory. It is aimed mathematicians without a background in logic. The goal is to provide the basic facts used for applications of descriptive set theory to other areas of mathematics,…
We explore "omitted label contexts," in which training data is limited to a subset of the possible labels. This setting is standard among specialized human experts or specific, focused studies. By studying Simpson's paradox, we observe that…
When we study the paradoxes of set theory we find out that there are mainly 2 types: the pathologies and the antinomies. These 2 notions are made precise and compared with the somehow inductively definable concept "abnormal". (See my paper…
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…
The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof…
A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the…
Increasing penetration of renewable energy introduces significant uncertainty into power systems. Traditional simulation-based verification methods may not be applicable due to the unknown-but-bounded feature of the uncertainty sets.…
We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…
The semantic paradoxes are associated with self-reference or referential circularity. However, there are infinitary versions of the paradoxes, such as Yablo's paradox, that do not involve this form of circularity. It remains an open…
This thesis presents the strong and weak forms of the Banach-Tarski paradox based on the Hausdorff paradox. It provides modernized proofs of the paradoxes and necessary properties of equidecomposable and paradoxical sets. The historical…
The structures $\langle M,\subseteq^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle…
The problem of comparing concepts of dependence in general rough sets with those in probability theory had been initiated by the present author in some of her recent papers. This problem relates to the identification of the limitations of…
We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…
This paper gives a counterexample to the impossibility, by G\"odel's second incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a fragment of arithmetic on the assumption that the latter is consistent.…
This paper provides some counterexamples to Cantor's contributions to the foundations of Set Theory. The first counterexample forces Cantor's Diagonal Method (DM) to yield one of the numbers in the target list. To study this anomaly, and…
As with a Bell inequality, Hardy's paradox manifests a contradiction between the prediction given by quantum theory and local-hidden variable theories. In this work, we give two generalizations of Hardy's arguments for manifesting such a…
Effective and reliable evaluation is essential for advancing empirical machine learning. However, the increasing accessibility of generalist models and the progress towards ever more complex, high-level tasks make systematic evaluation more…
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…
This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The…