Set Theory

Checkpoints

• Cantor's Theorem Basis Cantor's theorem proves that for any set A, its power set P(A) is always strictly larger — even when both are infinite. This checkpoint covers the diagonal argument behind the proof and the infinite tower of cardinalities that follows.
• Countable Set Basis An introduction to countable sets — explaining how bijections define cardinality for infinite sets, formally defining countability, and building intuition through explicit examples including the integers and rationals.
• Map Basis An introduction to maps between sets, covering domain, codomain, image, preimage, injectivity, surjectivity, bijectivity, composition, and inverse maps.
• Partial Order Basis Introduces partial orders — reflexive, antisymmetric, and transitive binary relations — covering key examples such as divisibility and set inclusion, Hasse diagrams, and the distinctions between minimal, maximal, least, and greatest elements.
• Power Set Basis An introduction to power sets — the set of all subsets of a given set — covering notation, explicit construction, and the key result that a set with n elements has exactly 2ⁿ subsets.
• Introduce to Set Basis A precise introduction to sets — the universal language of mathematics — covering notation, membership, subsets, core operations, and cardinality.
• Total Order Basis A total order extends partial order by requiring every pair of elements to be comparable, producing a single linear ranking with no incomparable pairs.
• Uncountable Set Basis A set is uncountable if it is infinite and admits no bijection with ℕ — no listing can exhaust it. This checkpoint proves ℝ is uncountable via Cantor's decimal diagonal argument, establishes |𝒫(ℕ)| = |ℝ| using the Cantor–Bernstein–Schroeder theorem, and places the continuum in the infinite hierarchy of cardinalities.
• Well-order Basis A well-order is a total order in which every non-empty subset has a least element — a property equivalent to mathematical induction on ℕ and connected to the Axiom of Choice for larger sets.
• ZFC Set Theory Basis A precise tour of the ten axioms of ZFC — the standard foundation of modern mathematics — explaining why naïve set theory fails, how each axiom fixes a specific gap, and how all of mathematics is built from the empty set up.