---
title: "Introduction to Metamathematics"
authors:
  - "Stephen C. Kleene"
year: 1952
venue: "North-Holland / Van Nostrand"
arxiv: null
doi: null
source: "https://openlibrary.org/works/OL5959470W/Introduction_to_metamathematics"
topics:
  - metamathematics
  - recursive-functions
  - kleene
  - textbook
seed_rank: 529
seed_batch: "foundations-greats-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 10
lineage: foundations-of-computing
cites:
  - title: "General recursive functions of natural numbers"
    url: "https://doi.org/10.1007/BF01565439"
    year: 1936
    arxiv: null
    doi: "10.1007/BF01565439"
  - title: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I"
    url: "https://doi.org/10.1007/BF01700692"
    year: 1931
    arxiv: null
    doi: "10.1007/BF01700692"
  - title: "Recursive predicates and quantifiers"
    url: "https://doi.org/10.1090/S0002-9947-1943-0007371-8"
    year: 1943
    arxiv: null
    doi: "10.1090/S0002-9947-1943-0007371-8"
  - title: "Investigations into Logical Deduction"
    url: "https://doi.org/10.1007/BF01201353"
    year: 1935
    arxiv: null
    doi: "10.1007/BF01201353"
see:
  - "525-general-recursive-functions-of-natural-numbers"
  - "500-uber-formal-unentscheidbare-satze-der-principia-mathematica-"
  - "528-recursive-predicates-and-quantifiers"
  - "632-investigations-into-logical-deduction"
---

# Introduction to Metamathematics

## One-sentence takeaway

Kleene's IMM is the mid-century encyclopaedia of recursion theory, proof theory, and intuitionistic systems—the book later computability texts still answer to.

## Why it matters here

Default cite for classical recursion theory in the Broadside archive. When Anoptic needs a single reference for primitive versus general recursion, Gödel numbering, or the arithmetical hierarchy, this is the one.

## Key ideas

- Systematic development of primitive and general recursive functions, including the normal-form and enumeration theorems.
- Full exposition of Gödel incompleteness in the Herbrand–Gödel–Kleene vocabulary.
- Surveys intuitionistic arithmetic and the relation of constructive to classical systems.
- First edition: North-Holland / Noordhoff / Van Nostrand, 1952, x+550 pp. (Bibliotheca Mathematica 1). No original ISBN; later reprints carry 0-7204-2103-9 / 0-444-10088-1.

## Caveats

## Links

- Open Library: https://openlibrary.org/works/OL5959470W/Introduction_to_metamathematics
- WorldCat 1952: https://search.worldcat.org/title/Introduction-to-metamathematics-by-Stephen-Cole-Kleene-../oclc/459805591