---
title: "On the Infinite"
authors:
  - "David Hilbert"
year: 1926
venue: "Mathematische Annalen (Über das Unendliche)"
arxiv: null
doi: "10.1007/BF01206605"
source: "https://doi.org/10.1007/BF01206605"
topics:
  - hilbert-program
  - finitism
  - infinity
seed_rank: 493
seed_batch: "foundations-greats-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 8
lineage: foundations-of-computing
cites:
  - title: "Mathematical Problems (Hilbert's Problems)"
    url: "https://www.ams.org/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf"
    year: 1900
    doi: "10.1090/S0002-9904-1902-00923-3"
see:
  - "487-mathematical-problems-hilbert-s-problems"
---

# On the Infinite

## One-sentence takeaway

Hilbert's Münster lecture (4 June 1925; *Math. Ann.* 95:161–190, 1926) defends infinitary mathematics as a system of ideal elements that must be secured by finitistic consistency proofs of the real propositions.

## Why it matters here

Pairs with incompleteness: this is the philosophical frame that says what a "finitary consistency proof" was supposed to mean before Gödel showed arithmetic cannot supply one.

## Key ideas

- Distinguishes real (finitary, contentual) propositions from ideal ones that mention completed infinities.
- "No one shall expel us from the paradise that Cantor has created" — but the paradise must be licensed by a finitary consistency proof.
- Finitary reasoning about concrete signs and proofs is the secure core; transfinite methods are instruments.
- Full *Annalen* text is the one to cite; a shorter version appeared in *Jahresbericht der DMV* 36 (1927).

## Caveats

## Links

- DOI: [10.1007/BF01206605](https://doi.org/10.1007/BF01206605)
- GDZ volume: https://gdz.sub.uni-goettingen.de/id/PPN235181684_0095