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7. Cantor's Theorem

This chapter discusses Cantor's theorem and the concept of cardinality in sets. It establishes that the cardinality of any set is strictly less than the cardinality of its power set, and provides various proofs, particularly using the diagonalization argument. The chapter concludes by revealing that there are infinitely many infinities, reflecting the nature of different cardinalities within infinite sets.

Sections

Cantor's Theorem

Cantor's Theorem states that the cardinality of any set is strictly less than the cardinality of its power set.

7 Section Overview

Start current section content and materials

7.1 Introduction to Cardinality

This section introduces the concept of cardinality and Cantor's theorem, highlighting the relationship between a set and its power set.

7.2 Proof by Contradiction

Cantor's theorem demonstrates that the cardinality of any set A is strictly less than the cardinality of its power set P(A), using proof by contradiction.

7.3 Construction of the Subset S

Cantor's theorem demonstrates that the cardinality of any set is strictly less than the cardinality of its power set, using a diagonalization argument.

7.4 Implications of Cantor's Theorem

Cantor's theorem demonstrates that the cardinality of any set is strictly less than the cardinality of its power set, establishing the existence of different sizes of infinity.

7.5 Conclusion

Cantor's theorem demonstrates that the cardinality of any set is strictly less than the cardinality of its power set, including infinite sets.

Learning Objectives

  • Cantor's theorem states that the cardinality of a set is strictly less than its power set.

  • The proof involves showing a contradiction through the diagonalization argument.

  • There are infinitely many different sizes of infinity, creating a hierarchy of cardinalities.

Key Concepts

Cantor's Theorem

A theorem stating that for any set A, the cardinality of A is strictly less than the cardinality of its power set P(A).

Cardinality

A measure of the 'number of elements' in a set, used to compare the sizes of different sets.

Diagonalization Argument

A proof technique used to show that certain sets are uncountable by constructing a subset that cannot be mapped to an existing set.

Power Set

The set of all subsets of a given set A, denoted as P(A).

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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