# Lowenheim-Skolem theorems

## Summary

The Lowenheim-Skolem theorems state that if a first-order theory has an infinite model, then it has models of every infinite cardinality. It implies that certain foundational theories in mathematics, such as Peano arithmetic and Zermelo-Frankl set theory, must have nonstandard models.

## Context

This concept has the prerequisites:

- semantics of first-order logic (The L-S theorems are about the semantics of first-order logic.)
- cardinality (The statements of the L-S theorems involve cardinality.)
- completeness of first-order logic (The downward L-S theorem follows from the proof of the Completeness Theorem.)
- compactness of first-order logic (The upward L-S theorem follows from the Compactness Theorem.)
- Zermelo-Frankl axioms (Skolem's Paradox is about Zermelo-Frankl set theory.)
- countable sets (Skolem's Paradox involves the diagonalization argument.)
- Peano axioms (One application of the L-S theorems is to show there are nonstandard models of the Peano axioms.)

## Goals

- Know the statements of the upward and downward L-S theorems.

- Prove the upward and downward L-S theorems.

- Know what Skolem's Paradox refers to and why it is paradoxical.

- Know why the L-S theorems imply nonstandard models of the Peano Axioms.

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

## -Free-

→ Notes on Logic (2013)

Lecture notes for a course on first order logic.

## -Paid-

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

- Section 5.1, "Basic ideas of model theory," pages 161-168
- Section 5.2, "The Lowenheim-Skolem Theorems," pages 168-173

→ A Mathematical Introduction to Logic

An undergraduate textbook in mathematical logic, with proofs.

Location:
Section 2.6, "Models of theories," subsection "Size of models," pages 151-155

## See also

-No Additional Notes-