Alphabeta Math

Set Theory

80 pages in 3 parts

Everything else in the library is a set, so this collection supplies the ground it stands on. The ZFC axioms come first, over a language whose only symbol is membership: extensionality, pairing, union, power set, separation, replacement, infinity, foundation and choice. The ordered pair, the Cartesian product, relations, functions, equivalence classes and quotients are then constructions rather than primitives. The natural numbers are built as the von Neumann finite ordinals, with induction and recursion proved instead of assumed. Partial orders, chains and maximal elements give Zorn's lemma, which is the form choice takes when the rest of the library reaches for it, and filters and ultrafilters are its first application. Well-ordering carries induction past the finite, transfinite recursion licenses definition by stages, and ordinal arithmetic, cardinals, cardinal arithmetic, cofinality and the alephs measure the sizes the library goes on to compare.

Every other collection here rests on this one, and several rest on named pages of it. The construction of the real numbers is a set-theoretic construction. Topology reaches for Zorn and the ultrafilter lemma at Tychonoff, at nets and filters, at the separation counterexamples and at the completion of a uniform space. Measure theory reserves transfinite recursion, cardinal arithmetic and cofinality, Zorn, quotients and ultrafilters. Probability reserves the relations and quotients page. Beyond what is built, forcing and the large-cardinal independence results start where the ordinals and cardinals here stop.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Sets, relations and functions

    2 pages

    Everything else in the library is a set, so the axioms come first: extensionality, pairing, union, power set, separation, replacement, infinity, foundation and choice, over a language whose only symbol is membership. On top of them the ordered pair, the Cartesian product, relations, functions and quotients are constructions rather than primitives, which is what lets a quotient later be taken without asking whether it exists.

    • The ZFC Axioms and the Basic Set Constructions39 results

      This development starts from first-order logic with equality over a single binary relation symbol ∈, in which every object of the domain is a set and the domain is nonempty; nothing mathematical is assumed before that.

      18 definitions, 7 lemmas, 3 propositions, 6 theorems, 3 corollaries, 2 remarksExamples & counterexamples →
    • Relations, Functions, and Quotients39 results

      The axioms and the basic constructions supply everything used here: the Kuratowski ordered pair and its characterising property, the Cartesian product built by Separation…

      15 definitions, 10 lemmas, 5 propositions, 5 theorems, 3 corollaries, 1 remarkExamples & counterexamples →

  2. Part 2 · The naturals, order and choice

    5 pages · after Part 1

    The natural numbers are built as the von Neumann finite ordinals, with induction and recursion proved rather than assumed. Formal syntax for arbitrary set signatures makes this recursion precise: parsing supports term denotation, satisfaction, substitution, renaming, isomorphism, and relativization within ZF. Partial orders, chains, and maximal elements give Zorn's lemma, where full choice enters. The dependent-choice page works over ZF, reconciles the serial-relation and category formulations, and proves equivalence with the complete-metric Baire principle while identifying local uses of choice. Filters and ultrafilters are the first application of full choice: a maximal filter exists because Zorn says so.

  3. Part 3 · Ordinals and cardinals

    34 pages · after Part 2

    Ordinals: recursion, rank, arithmetic, cofinality, alephs, clubs, Diamond, determinacy, PCF, completeness, incompleteness, reflection, BPI, Stone duality, Halpern--Läuchli. Large cardinals: ultrafilters, ultrapowers, supercompactness yielding PFA, Prikry forcing, Gitik's all-singular model; Suslin trees, L, HOD, GCH. Forcing: generics, names, truth lemma, chain conditions, Cohen and Lévy collapse, finite-support MA+¬CH, symmetric extensions, permutation models, Easton class forcing realizing monotone continuum values at infinite regular cardinals under the cofinality constraint, singular cardinals excluded. Invariants: p,t,b,d,s,r with the null and meagre cardinals, the Cichoń diagram. Choice strength: complete-metric Baire = DC, compact-Hausdorff Baire = DMC, Urysohn from DMC but not CC or BPI, Stone metrization failing under DC and BPI, the normal Moore space conjecture proved under PMEA and refuted by V=L, Shelah's equiconsistent model.