Mathematics / research
Axiomatic set theory and independence
Understand mathematical infinity through axioms, ordinal recursion, cardinal arithmetic, constructibility, and forcing.
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Module 01 research
Axiomatic set theory and independence
Understand mathematical infinity through axioms, ordinal recursion, cardinal arithmetic, constructibility, and forcing.
- 01
Zermelo–Fraenkel axioms and Replacement
Distinguish bounded comprehension from functional collection and track the axioms that make a mathematical construction a set.
33 min - 02
Ordinals and transfinite induction
Use well-ordered transitive sets as order types and distinguish ordinal addition from cardinal counting.
32 min - 03
Transfinite recursion and the rank hierarchy
Construct set-length histories, distinguish definable class recursion from set-valued functions, and locate sets in the cumulative hierarchy.
34 min - 04
Cardinals and the Axiom of Choice
Distinguish bijective size from order type and identify exactly where well-ordering and simultaneous selections use Choice.
33 min - 05
Cofinality and cardinal arithmetic
Track unbounded sequences, regular and singular cardinals, and diagonal restrictions on exponentiation under ZFC.
35 min - 06
Stationary sets and club filters
Use regular uncountable cardinals to distinguish unboundedness, closed-unbounded sets, and stationary intersection principles.
35 min - 07
The constructible universe and relative consistency
Define Gödel’s hierarchy using set-sized definability and distinguish a class interpretation from an asserted set model of ZFC.
37 min - 08
Forcing partial orders and generic filters
Build compatible approximations, meet ground-model dense sets, and keep external countability distinct from internal cardinality.
36 min - 09
Forcing names and the forcing theorem
Interpret recursively coded names, distinguish truth in a generic extension from a condition’s forcing assertion, and state the theorem as a formula scheme.
38 min - 10
The Continuum Hypothesis and relative consistency
Combine constructibility, Cohen forcing, cardinal preservation, and name counting without confusing a conditional independence theorem with a proof of consistency.
40 min