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-