Cardinals Summary

Set Theory · Axiom Academy

Let's review the key concepts about comparing infinite sets and the hierarchy of infinities. Cardinality: Two sets have the same cardinality if there exists a bijection between them Bijections as the Key: A one-to-one and onto function proves sets have equal "size" Beyond Counting: Cardinality extends the idea of "how many" to infinite sets Why It Matters: Allows rigorous comparison of infinite collections The Countable-Uncountable Divide Countable Sets: Sets with cardinality less than or equal to Smallest Infinity: Natural numbers have cardinality , the smallest infinite cardinal Surprising Countability: Integers, rationals, and algebraic numbers are all countable Uncountable Sets: Real numbers, power set of naturals—infinities too large to list Cantor's Revolutionary Discoveries Not All Infinities Are Equal: Cantor proved that the real numbers are uncountable, showing there are different sizes of infinity Diagonal Argument: The ingenious technique showing cannot be put in one-to-one correspondence with Cantor's Theorem: For any set A, the power set has strictly greater cardinality than A Infinite Hierarchy: This implies infinitely many different sizes of infinity: Schroder-Bernstein Theorem: If and , then Cantor's Theorem: for all sets A—no largest cardinal Cardinal Arithmetic: For infinite cardinals, Continuum: is the cardinality of the reals Countable Operations: Countable union of countable sets is countable Products and Exponents: reveals structure of infinite cardinals

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