---
tag: Q24EQE2HVG
kind: definition
name: Foundations of mathematics
msc: [03A05, 03B30]
defines: EFFRJFW67Y
uses: [086QKPF0A6, 0B94TKFQJX]
sources:
  - url: https://en.wikipedia.org/wiki/Foundations_of_mathematics
---

## Definition

The *foundations of mathematics* is the part of mathematical logic and philosophy that studies the basic framework of mathematics.
It studies:

1. the formal languages in which mathematical statements are written, such as [[predicate logic|086QKPF0A6]];
2. the axiom systems from which mathematical theorems are derived, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]];
3. the notion of [[proof|0B94TKFQJX]], and the properties of the whole framework, such as its consistency and its completeness.

## Remarks

For example, the [[Peano axioms|0QFT7Z76N5]] can be interpreted in set theory.
Each natural number becomes a set, and each axiom becomes a theorem of set theory.
A theorem about prime numbers alone, such as the infinitude of primes, is not a foundational result: it is about objects inside the framework, not about the framework.
The main schools of the early twentieth century were [[logicism|5PGSQYB5Y1]], [[formalism|46JCQ6KDA4]] and [[intuitionism|XQHF63BBYF]].