Related papers: Set-theoretic mereology
Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative…
This paper obtains a completeness result for inequational reasoning with applicative terms without variables in a setting where the intended semantic models are the full structures, the full type hierarchies over preorders for the base…
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…
We formulate the Hauptvermutung of Causal Set Theory in two mathematically well-defined but different ways one of which turns out to be wrong and the other one turns out to be true. A further result is that the Hauptvermutung is true if we…
Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general…
Justification theory is a unifying framework for semantics of non-monotonic logics. It is built on the notion of a justification, which intuitively is a graph that explains the truth value of certain facts in a structure. Knowledge…
We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational…
A question is proposed whether or not set theory is consistent.
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other…
This paper provides a model theoretic semantics to feature terms augmented with set descriptions. We provide constraints to specify HPSG style set descriptions, fixed cardinality set descriptions, set-membership constraints, restricted…
The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine…
This article will be a continuation of our research into self-justifying systems. It will introduce several new theorems and their applications. (One of these results will transform our previous infinite-sized self-verifying formalisms into…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
We acuminate the idea of a final theory of physics in order to analyze its logical implications and consequences. It is argued that the rationale of a final theory is the principle of sufficient reason. This implies that a final theory of…
The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…