---
title: "Investigations in the foundations of set theory I"
authors:
  - "Ernst Zermelo"
year: 1908
venue: "Mathematische Annalen (Untersuchungen über die Grundlagen der Mengenlehre I)"
arxiv: null
doi: "10.1007/BF01449999"
source: "https://doi.org/10.1007/BF01449999"
topics:
  - set-theory
  - zermelo
  - axioms
seed_rank: 495
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"
---

# Investigations in the foundations of set theory I

## One-sentence takeaway

Zermelo's 1908 axiomatization replaces the naive comprehension that produced Russell's paradox with a short list of axioms — extensionality, elementary sets, separation, power set, union, choice, infinity — so set theory becomes a formal theory.

## Why it matters here

ZF/ZFC is the ambient universe when Anoptic talks about mathematical semantics and inductive definitions; this is the first published axiom list.

## Key ideas

- *Aussonderung* (separation): only subsets of an already given set defined by a definite property exist, blocking the Russell set.
- Choice is stated explicitly as an axiom, not a theorem — the paper that made the axiom of choice a named hypothesis.
- Infinity guarantees an inductive set; together with the others this is the core of later ZF (Fraenkel/Skolem replacement and regularity come later).
- *Math. Ann.* 65:261–281; DOI 10.1007/BF01449999.

## Caveats

## Links

- DOI: [10.1007/BF01449999](https://doi.org/10.1007/BF01449999)