# ultrafilters (under construction)

## Summary

-No Summary-

## Notes

This concept is still under construction.

## Context

This concept has the prerequisites:

- Zorn's Lemma (Zorn's Lemma is needed to show that every filter is contained in an ultrafilter.)
- set operations (Ultrafilters are defined on power sets.)
- Boolean algebras (Ultrafilters can be defined on Boolean algebras.)

## Goals

- Define filter

- Define ultrafilter

- Define principal ultrafilter

- Give an example of a non-principal ultrafilter

- Using Zorn's Lemma, show that every filter is contained in an ultrafilter

- Show that in the case of Boolean algebras, ultrafilters are equivalent to homomorphisms to {0, 1}.

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

## -Free-

→ Notes on Logic (2013)

Lecture notes for a course on first order logic.

Location:
Section 13, "Filters and ultrafilters," pages 30-33

## -Paid-

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

Location:
Section 4.3, "Filters and homomorphisms," pages 133-141

## See also

-No Additional Notes-