Frege’s Basic Law V and the Foundations of Arithmetic

Overview

Basic Law V, a cornerstone of Gottlob Frege’s logicist project, proposed a way to define concepts based on the objects they encompass. This principle was intended to provide a rigorous logical foundation for arithmetic.

Key Concepts

The core idea of Basic Law V is that the extension of a concept is identical to the extension of another concept if and only if they refer to the same objects. This is often expressed as the ‘value-range’ of a function.

Deep Dive

Frege’s formalization of this idea, often written as:

(εξFξ = εGε) ↔ (∀F)(∀G)(F=G)

This law allows for the identification of sets (or extensions) based on their members. While powerful, it proved to be the source of significant paradoxes.

Applications

The primary application was to define numbers. For example, the number 2 could be defined as the extension of the concept ‘equinumerous with the concept of being a planet in our solar system’.

Challenges & Misconceptions

The most significant challenge came from Bertrand Russell, who demonstrated that Basic Law V leads to a contradiction, now known as Russell’s Paradox. This paradox undermined Frege’s entire logicist program.

FAQs

What is Russell’s Paradox? It arises from considering the set of all sets that do not contain themselves. If such a set contains itself, it shouldn’t; if it doesn’t contain itself, it should. This contradiction invalidated Basic Law V.