Date
12-15-2025
Document Type
Thesis
Abstract
The compactness theorem for first-order logic says that for a set of first-order sen-tences to be satisfiable, it’s enough for every finite subset to be satisfiable. We’ll see how the compactness theorem arises from two methods of model construction: first by ultraproducts, and second by Henkin’s method of constructing models from constant symbols. We’ll explore several representative applications of compactness, such as how it can be used to demonstrate that a property is not expressible, establishing limitations on first-order expressibility. The Löwenheim-Skolem theorem says that if an infinite set Γ of first-order sentences has an infinite model, then for all κ ≥ |Γ| there is a model of Γ with cardinality κ. Like the compactness theorem, this result also places bounds on expressibility, namely the expressibility of infinite cardi-nalities. Some classical consequences we’ll see include Skolem’s paradox and a sufficient condition for a theory to be complete. Finally we’ll investigate analogues to the compactness and Löwenheim-Skolem theorems in other logics, such as classical and intuitionistic propositional logic, fuzzy logic, classical second-order logic, and the infinitary logic Lω1,ω.
Department
Mathematics
Thesis Committee
Dr. Ward Heilman, Thesis Advisor
Dr. Rachel Stahl, Committee Member
Dr. Stephen Flood, Committee Member
Copyright and Permissions
Original document was submitted as an Honors Program requirement. Copyright is held by the author.
Recommended Citation
Murphy, Adam. (2025). The Mechanics and Consequences of the Löwenheim-Skolem and Compactness Theorems. In BSU Honors Program Theses and Projects. Item 720. Available at: https://vc.bridgew.edu/honors_proj/720
Copyright © 2025 Adam Murphy