This is the TOC and Bibliography for my posts “Set Theory Jottings”
Posts
Naive Set Theory
Set Theory Jottings 1. Philosophy and Naive Set Theory: Formalism, Platonism, Intuitionism. Also, naive vs. axiomatic.
Set Theory Jottings 2. Cantor’s Paradise: A potted history of Cantor’s mathematical journey. And some myths I used to believe about it.
Set Theory Jottings 3. The Paradoxes: The Largest Cardinal, The Largest Ordinal, and Russell’s. Also, Cantor and the adage, “Never let a crisis go to waste.”
Set Theory Jottings 4. Ordinals: Cantor’s definitions, and how they evolved. The modern (von Neumann) definition. Picturing countable ordinals. The cardinality ℵ1.
Set Theory Jottings 5. Zermelo to the Rescue! (Part 1): Zermelo’s Axiom of Choice and his proof of the Well-Ordering Theorem. Objections to it, and Zermelo’s replies.
Set Theory Jottings 6. Zorn’s Lemma: Proofs using Zorn’s Lemma compared with proofs using transfinite induction.
Set Theory Jottings 7. The (Cantor-Dedekind-Schröder)-Bernstein Theorem: If 𝔪≤𝔫 and 𝔫≤𝔪, then 𝔪=𝔫. Also: a version for order types.
Set Theory Jottings 8. Ordinal Arithmetic: Direct definitions, and definitions via transfinite induction. Equalities and inequalities.
Set Theory Jottings 9. Cantor Normal Form: Kinda like polynomials. Or decimal notation.
Axiomatic Set Theory
Set Theory Jottings 10. Axiomatic Set Theory: Generalities, including: do the axioms have to be “true”? Viewpoints: the Greeks, Cantor, Dedekind, Frege, Russell, and Hilbert.
Set Theory Jottings 11. Zermelo to the Rescue! (Part 2): Zermelo’s paper “Investigations in the foundations of set theory”.
Set Theory Jottings 12. Zermelo on “definiteness”: Zermelo’s paper “On the concept of definiteness in axiomatics”.
Set Theory Jottings 13. From Zermelo to ZFC: Everything’s a Set! ZFC is a “pure” set theory. Kuratowski’s ordered pairs, von Neumann’s ordinals and cardinals. Definitions by transfinite recursion.
Set Theory Jottings 14. From Zermelo to ZFC: Replacement and Foundation: Two more axioms to add to the list. The cumulative hierarchy, ranks, transitive closure, and ∈-induction.
Set Theory Jottings 15. From Zermelo to ZFC: Formal Logic: Skolem’s proposal: “definite” means “first-order definable”. Coda: proper classes and NBG.
Set Theory Jottings 16. Axioms of ZFC: The axioms, at last.
Set Theory Jottings 17. Ordinals Revisited: Some fine points, including proof by transfinite induction and definition by transfinite recursion.
Set Theory Jottings 18. The Axiom of Determinacy: An alternative to the Axiom of Choice.
Set Theory Jottings 19. GCH implies AC: Sierpiński’s Theorem.
Consistency of GCH and AC
Set Theory Jottings 20. Consistency of GCH and AC: Overview: Five foundation stones for Gödel’s proof of this.
Bibliography
John Baez. “Large Countable Ordinals”. Part 1; Part 2; Part 3
Paul J. Cohen. Set Theory and the Continuum Hypothesis. W.A. Benjamin, 1966.
Joseph Dauben. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press, 1979.
Heinz-Dieter Ebbinghaus. Ernst Zermelo: An Approach to His Life and Work. 2nd edition, 2015, Springer.
Paul Erdős. “An interpolation problem associated with the continuum hypothesis”. Michigan Mathematics Journal, pp.9–10, vol.11 no.3 1964.
Irving Kaplansky. Set Theory and Metric Spaces. Allyn and Bacon, 1972.
David Madore. An Applet to Draw Ordinals.
Gregory Moore. “The Origins of Zermelo’s Axiomatization of Set Theory”. Journal of Philosophical Logic. 7:1, January 1978, pp.307–329.
Gregory Moore and Alejandro Garciadiego. “Burali-Forti’s Paradox: A Reappraisal of its Origins”. Historia Mathematica, August 1981 (8:3), pp.319–350.
Gregory Moore. Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. Springer, 1982.
Joseph G. Rosenstein. Linear Ordering. Academic Press, 1982.
John Stillwell. Roads to Infinity. A K Peters, 2010
Diagonal Argument 