---
title: "Systems of Logic Based on Ordinals"
authors:
  - "Alan M. Turing"
year: 1939
venue: "Proceedings of the London Mathematical Society"
arxiv: null
doi: "10.1112/plms/s2-45.1.161"
source: "https://doi.org/10.1112/plms/s2-45.1.161"
topics:
  - ordinal-logics
  - turing
  - proof-theory
  - incompleteness
seed_rank: 518
seed_batch: "foundations-greats-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 10
lineage: foundations-of-computing
cites:
  - title: "On Computable Numbers, with an Application to the Entscheidungsproblem"
    url: "https://doi.org/10.1112/plms/s2-42.1.230"
    year: 1936
    arxiv: null
    doi: "10.1112/plms/s2-42.1.230"
  - 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: "On notation for ordinal numbers"
    url: "https://doi.org/10.2307/2267778"
    year: 1938
    arxiv: null
    doi: "10.2307/2267778"
  - title: "A Formulation of the Simple Theory of Types"
    url: "https://doi.org/10.2307/2266170"
    year: 1940
    arxiv: null
    doi: "10.2307/2266170"
see:
  - "515-on-computable-numbers-with-an-application-to-the-entscheidun"
  - "500-uber-formal-unentscheidbare-satze-der-principia-mathematica-"
  - "527-on-notation-for-ordinal-numbers"
  - "510-a-formulation-of-the-simple-theory-of-types"
---

# Systems of Logic Based on Ordinals

## One-sentence takeaway

Turing's Princeton thesis builds ordinal logics: a transfinite progression of theories, each adjoining the consistency statement of its predecessor, indexed by Kleene's notations for recursive ordinals.

## Why it matters here

Incompleteness is not a wall but a climb. Any Anoptic story about self-certifying systems, proof strength, or "add the next axiom and continue" is downstream of these O-logics and of the oracle machines (o-machines) introduced along the way.

## Key ideas

- An ordinal logic is a recursive assignment of formal systems to notations in Kleene's \(\mathcal{O}\).
- Each successor step adds a consistency (or reflection) statement; limit steps take unions along a notation.
- Completeness fails in general, but the construction isolates how much incompleteness can be climbed past at each recursive ordinal.
- Introduces oracle machines as a tool for relative computability—the ancestor of Turing reducibility.

## Caveats

## Links

- DOI: [10.1112/plms/s2-45.1.161](https://doi.org/10.1112/plms/s2-45.1.161)
- URL: https://doi.org/10.1112/plms/s2-45.1.161