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Mathematics
Mathematical Logic
Set Theory
Axiomatic Set Theory
Foundations of Set Theory
Naive Set Theory
Limitations and Paradoxes
Russell's Paradox
Cantor's Paradox
Zermelo-Fraenkel Set Theory (ZF)
Axioms of ZF
Axiom of Extensionality
Axiom of Regularity (Foundation)
Axiom of Pairing
Axiom of Union
Axiom of Power Set
Axiom of Infinity
Axiom of Replacement
Axiom of Empty Set
Axiom Schema of Separation
Role in Avoiding Paradoxes
Axiom of Choice (AC)
Various Formulations and Equivalents
Zorn's Lemma
Well-Ordering Theorem
Implications and Controversies
Applications in Mathematics
Continuum Hypothesis (CH)
Statement and History
Relationship with ZF and AC
Independence from ZF
Ordinals and Cardinals
Ordinal Numbers
Definition and Construction
Well-ordered Sets
Transfinite Induction and Recursion
Arithmetic of Ordinals
Ordinal Addition, Multiplication, Exponentiation
Limit Ordinals and Successor Ordinals
Cardinal Numbers
Cardinality of Sets
Definition and Properties
Cardinal Arithmetic
Cardinal Addition, Multiplication, Exponentiation
Infinite Cardinals
Aleph Numbers
Comparison with Continuum
Independence Results
Gödel's Constructible Universe (L)
Construction and Properties
Role in Proving Consistency
Cohen's Method of Forcing
Basic Concepts and Techniques
Applications in Proving Independence
Examples: CH and AC
Large Cardinals
Concept and Motivation
Hierarchies of Large Cardinals
Measurable Cardinals
Inaccessible Cardinals
Strong and Supercompact Cardinals
Impact on Set Theory and Mathematics
Consistency Strength
Role in Exploring Foundations of Mathematics
3. Model Theory
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5. Computability Theory