Lambda Calculus Synopsis

TL;DR

This paper provides a concise overview of key concepts, notations, and propositions from Barendregt's Lambda Calculus, including syntax, semantics, combinatory logic, fixed point theorem, and a topological perspective.

Contribution

It offers a simplified, linear summary of fundamental lambda calculus topics and introduces a topology to explain the apparent contradiction in the set of lambda terms.

Findings

Defined a theory of untyped extensional lambda calculus

Described a topology on lambda terms to explain continuity

Summarized key propositions and notations from Barendregt's monograph

Abstract

This text gives a rough, but linear summary covering some key definitions, notations, and propositions from Lambda Calculus: Its Syntax and Semantics, the classical monograph by Barendregt. First, we define a theory of untyped extensional lambda calculus. Then, some syntactic sugar, a system of combinatory logic, and the fixed point theorem are described. The final section introduces a topology on the set of lambda terms which is meant to explain an illusory contradiction. Namely, functions defined on the set of lambda terms are in the set of lambda terms itself, the latter being a countable set. However, the functions on the set of lambda terms appear to be continuous with respect to a topology of trees.