---
tag: EFFRJFW67Y
kind: concept
name: Foundations of mathematics
msc: [03A05, 03B30]
sources:
  - url: https://en.wikipedia.org/wiki/Foundations_of_mathematics
same_as:
  wikidata: Q833585
  wikipedia: Foundations of mathematics
---

## Summary

The foundations of mathematics study the basic framework in which all of mathematics is built.
They ask which language mathematics is written in, which axioms it starts from, and what counts as a [[proof|0B94TKFQJX]].
Today the usual answer is [[predicate logic|086QKPF0A6]] together with the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
The field also studies the limits of this framework, such as the [[incompleteness theorems|X9WYNYE29C]].

## Explanations

### L1 human

Every theorem rests on earlier theorems, and these rest on definitions and axioms.
The foundations of mathematics look at the bottom of this tower.
For example, the natural numbers can be built from sets: $0$ is the empty set, $1$ is $\set{0}$, $2$ is $\set{0, 1}$, and so on.
Then facts about numbers become facts about sets, and only the axioms of set theory are needed.
Around 1900, paradoxes in naive set theory made this question urgent, and [[logicism|5PGSQYB5Y1]], [[formalism|46JCQ6KDA4]] and [[intuitionism|XQHF63BBYF]] gave different answers.