14 research outputs found
Completeness and Categoricty, Part II: 20th Century Metalogic to 21st Century Semantics
This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic
Completeness and Categoricty, Part II: 20th Century Metalogic to 21st Century Semantics
This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic
Completeness and Categoricity, Part I: 19th Century Axiomatics to 20th Century Metalogic
This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic
"Clarifying the Nature of the Infinite": the Development of Metamathematics and Proof Theory
We discuss the development of metamathematics in the Hilbert school,
and Hilbert’s proof-theoretic program in particular. We place this program
in a broader historical and philosophical context, especially with
respect to nineteenth century developments in mathematics and logic.
Finally, we show how these considerations help frame our understanding
of metamathematics and proof theory today
Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics
Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics