User:Jdapayne/Logicist constructions of arithmetic

This article is about various related logicist constructions of arithmetic which have been associated with the logicist programmed of Gottlob Frege and Bertrand Russell. They have taken the form of a derivation of the basic laws of arithmetic from principles which are claimed to be purely logical.

Frege's construction

Frege's construction spans three books - his Begriffsschrift, in which he develops a system of formal logic. In The Foundations of Arithmetic he informally sketches an extension to this system for the purposes of constructing arithmetic. It is also mainly in this book in which he sets forth the philosophy underpinning the system. Finally, in The Basic Laws of Arithmetic he develops a formal system roughly corresponding to the informal system in Foundations.

Logical preliminaries

Begriffschrift, Concept and object, second-order logic

Hume's principle

=== Frege's final definition and basic Law v ===

Including derivation of HP?

Russell's paradox

However, Frege's system faced a fatal flaw. It turns out that Frege's theory of classes as embodied in BLV is inconsistent. This fact was communicated to Frege in a letter from Russell.

The problem is the following: In Frege's system, there is a concept corresponding to every formula of the language. And then, as a consequence of blv, there is a class corresponding to every concept. Then, consider the concept (finish)

Russell's construction

Copy and paste from logic is, article