---
title: "What is Cantor's Continuum Problem?"
authors:
  - "Kurt Gödel"
year: 1947
venue: "American Mathematical Monthly"
arxiv: null
doi: "10.2307/2305611"
source: "https://doi.org/10.2307/2305611"
topics:
  - continuum-hypothesis
  - godel
  - set-theory
seed_rank: 502
seed_batch: "foundations-greats-2026-08-13"
reviewed: "2026-08-13"
pool: "maths-foundations"
relevance_score: 8
lineage: foundations-of-computing
cites:
  - title: "The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory"
    url: "https://archive.org/details/consistencyofaxi0054gode"
    year: 1940
    arxiv: null
    doi: null
  - title: "Mathematical Problems (Hilbert's Problems)"
    url: "https://mathcs.clarku.edu/~djoyce/hilbert/problems.html"
    year: 1900
    arxiv: null
    doi: null
  - title: "Investigations in the foundations of set theory I"
    url: "https://doi.org/10.1007/BF01449999"
    year: 1908
    arxiv: null
    doi: "10.1007/BF01449999"
see:
  - "501-the-consistency-of-the-axiom-of-choice-and-of-the-generalize"
  - "487-mathematical-problems-hilbert-s-problems"
  - "495-investigations-in-the-foundations-of-set-theory-i"
---

# What is Cantor's Continuum Problem?

## One-sentence takeaway

Gödel treats the continuum hypothesis as a definite mathematical question that current axioms leave open, and predicts that new axioms of infinity may settle it.

## Why it matters here

Broadside's archive spine is full of independence and completeness results; this essay is the tone-setting argument that an undecided question is still a scientific one. When Anoptic debates whether a standing rule, type, or simulation axiom is "just convention," Gödel's reply is: keep looking for the next axiom that forces a unique answer.

## Key ideas

- After the 1940 relative-consistency proof, CH is not refutable from ZF + AC, but Gödel refuses to read that as "CH has no meaning."
- He argues that set theory describes an objective universe of sets, so \(2^{\aleph_0} = \aleph_1\) is either true or false there.
- The essay sketches why \(V = L\) looks too thin to be the whole universe and why large-cardinal axioms might decide CH.
- The 1964 revised version (in Benacerraf–Putnam) restates the same program after Cohen's independence result.

## Caveats

## Links

- DOI: [10.2307/2305611](https://doi.org/10.2307/2305611)
- JSTOR: https://www.jstor.org/stable/2305611