# incompleteness of set theory

## Summary

Godel's Incompleteness Theorems apply to any formal deductive system which includes arithmetic. Since the natural numbers can be defined using set theory, the theorems apply to set theory as well.

## Context

This concept has the prerequisites:

- Zermelo-Frankl axioms (One must choose a set of axioms to show the incompleteness of.)
- Godel's Incompleteness Theorems (The result is derived as a consequence of Godel's Incompleteness Theorems.)
- natural numbers as sets (The interpretation from arithmetic into set theory requires defining natural numbers as sets.)
- interpretations between theories (Extending Godel's incompleteness results to set theory requires an interpretation of arithmetic into set theory.)

## Goals

- Extend Godel's Incompleteness Theorems (stated in terms of arithmetic) to axiomatic set theory

## Core resources (read/watch one of the following)

## -Paid-

→ A Mathematical Introduction to Logic

An undergraduate textbook in mathematical logic, with proofs.

Location:
Section 3.7, "Second Incompleteness Theorem," starting with "Applications to set theory," pages 270-275

## See also

-No Additional Notes-