arXiv subjects

Alec Rhea

Publications and source records attributed to Alec Rhea.

4 recordsLinked to original sources

arXiv2022

An axiomatic approach to higher order set theory

Higher order set theory has been a topic of interest for some time, with recent efforts focused on the strength of second order set theories [KW16]. In this paper we strive to present one 'theory of collections' that allows for a formal consideration of 'countable higher order set theory'. We will see that this theory is equiconsistent with $ZFC$ plus the existence of a countable collection of inaccessible cardinals. We will also see that this theory serves as a canonical foundation for some parts of mathematics not covered by standard set/class theories (e.g. $ZFC$ or $MK$), such as category theory.

math.LO↗

arXiv2022

An axiomatic approach to the multiverse of sets

Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a simultaneous consideration of all set theoretical universes and the relationships between them, eliminating the need for recourse 'outside the theory' when carrying out constructions like forcing etc. We also explore multiversal category theory, showing that we are finally free of questions about 'largeness' at each stage of the categorification process when working in this theory -- the category of categories we consider for a given universe contains all large categories in that universe without taking recourse to a larger universe. We leverage this newfound freedom to define a category ${\bf Force}$ whose objects are universes and whose arrows are forcing extensions, a $2$-category $\mathcal{V}\mathfrak{erse}$ whose objects are the categories of sets in each universe and whose component categories are given by functor categories between these categories of sets, and a tricategory $\mathbb{Cat}$ whose objects are the $2$-categories of categories in each universe and whose component bicategories are given by pseudofunctor, pseudonatural transformation and modification bicategories between these $2$-categories of categories in each universe.

math.LO↗

arXiv2017

Transfinite Galois Theory

In this paper I generalize the notion of a polynomial over an ordered field to that of a naked polynomial over a non-Archimedean ordered field, subsequently showing that the notion of a naked polynomial ring forms an Euclidean domain. This canonically generalizes the methods of Galois theory of fields and polynomial rings to a transfinite Galois theory of non-Archimedean ordered fields and naked polynomial rings, lifting the processes of splitting and algebraic closure to non-Archimedean ordered fields.

math.RA↗

arXiv2017

The Ordinals as a Consummate Abstraction of Number Systems

In the course of many mathematical developments involving 'number systems' like $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb {C}$ etc., it sometimes becomes necessary to abstract away and study certain properties of the number system in question so that we may better understand objects having these properties in a more general setting -- a relevant example for this paper would be Hausdorffs $η_\varepsilon$-fields, objects abstracted from the property that there are no two disjoint intervals in $\mathbb{R}$ of cardinality $α\leq \aleph_0$ whose union is $\mathbb{R}$. My goal will be to reverse this process, so to speak -- I will begin in one of the most abstract mathematical settings possible, using only the undefined notions and axioms of MK class theory to define and subsequently add structure to the class of all ordinals, $O_n$. I proceed in this fashion until the construction (or a subclass thereof) is structurally isomorphic to whichever 'number system' we wish to consider. In the course of doing so, I construct a proper class of set-sized non-isomorphic discretely ordered rings and a proper class-sized discretely ordered ring, a proper class of non-isomorphic set-sized densely ordered fields that are not real-closed and a proper class sized densely ordered field that is not real-closed, a proper class of non-isomorphic real-closed fields, and a proper class of non-isomorphic algebraically closed fields. I also propose a new definition for the Surreal and Surcomplex numbers, using a 'final culmination' of the process used to construct real-closed fields and their algebraic closures. This process yields a straightforward and logically satisfying method for moving back and forth between well-studied number systems like $\mathbb{R}$ and very abstract number systems, such as an $η_{_{ω^ω}}$-field.

math.LO↗