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Set Theory

Class at Faculty of Education |


Antinomies of Cantor's intuitive set theory.

Comparison of sets. Sets of the same cardinality. Finite and infinite sets. Constructive and existential proofs of equivalence of two infinite sets. Cantor's theorem about the cardinality of the power set.

Countable and uncountable sets. Union and Cartesian product of two countable sets, the proof of their countability. The union and Cartesian product of countable many countable sets and the question of their countability. Uncountable sets and sets of the cardinality of continuum. The uncountability of the set of all real numbers. Sets of higher cardinalities.

Cantor's discontinuum. Uncountability of the discontinuum. The equivalence of a square with its side. cardinal numbers and operations with them. Comparison of the arithmetic of cardinal numbers witrh the arithmetic of finite numbers. Alephs. The continuum hypothesis.

Ordered sets and well ordering. Ordinal numbers and their arithmetic. The non-commutativity of the arithmetic operations with infinite ordinals. Limit ordinals. The princile of transfinite induction.

The axiom of choice, its consequences and its alternatives. The well ordering of any set.


Elements of the set theory. Cardinality of a set, countable and uncountable sets.

Cardinal and ordinal numbers, Zermelo's axiom and its consequences. Cantor discontinuum and its properties.

Peano's curve.