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**Topics for Sets Theory**

abstract mathematical thinking, basic definitions, relations, functions, equivalence relations, inductive sets, cardinality of sets, Logic, propositional calculus, first-order logic, semantics, proof systems, soundness, completeness theorems, compactness theorem, philosophical overtones, paradoxes, Russell, Cantor, Burali-Forti, motivating axiomatic, Zermelo-Fraenkel, axioms, axiom schemata, Transfinite induction, recursion, cardinal numbers, ordinal numbers, real number system , Zermelo-Fraenkel set theory, Continuum Hypothesis, Inaccessible Cardinals,

Paradoxes

The cumulative hierarchy

The axioms for Zermelo-Fraenkel's set theory

Classes

Ordered sets: partial and linear orderings

well-founded relations

well-orderings

The axiom of choice and equivalent variants

Zorn's lemma and the well-ordering principle

Transfinite induction and recursion.

Ordinals and cardinals

The continuum hypothesis

independence results and models of set theories

category theory

Set Theory & Mathematical Logic

the Zermelo-Fraenkel Axioms

Axiom of Choice

Transfinite arithmetic

Zorn's Lemma

Ordinal numbers

Cardinal numbers

model theory

Propositional & predicate calculus

semantics

soundness & completeness of formal languages

recursive functions & computability

Godel's incompleteness theorems

The axioms of set theory

The axioms of ZFC

The theory of transfinite ordinals and cardinals

Cardinals

The universe VV of all sets

Set theory as the foundation of mathematics

Metamathematics

The incompleteness phenomenon

The set theory of the continuum

Descriptive Set Theory

Determinacy

The Continuum Hypothesis

Gödel's constructible universe

Forcing

Other applications of forcing

The search for new axioms

Large cardinals

Inner models of large cardinals

Consequences of large cardinals

Forcing axioms