Uncountable Sets

Set Theory · Axiom Academy

Discovering infinities larger than the natural numbers Georg Cantor proved that the real numbers are uncountable using a brilliant proof by contradiction. Assume we could list all real numbers in (0,1). We construct a new number by changing the n -th digit of the n -th number in our list. This new number differs from every number in our supposed complete list in at least one digit, creating a contradiction. Therefore, no such enumeration exists, and the real numbers are uncountable. 2. The Cardinality of the Continuum The cardinality of the real numbers is denoted by 𝔠 (pronounced "continuum"). This represents a fundamentally different size of infinity than ℵ₀, the cardinality of countable sets. We can prove that 𝔠 = 2^ℵ₀, showing that the continuum has the same cardinality as the power set of the natural numbers. This establishes a hierarchy of infinities: countable sets are strictly smaller than the continuum. 3. The Power Set of Natural Numbers The power set of the natural numbers, P(ℕ), is the set of all subsets of ℕ. Remarkably, this set is uncountable and has the same cardinality as the real numbers. We can establish a bijection between P(ℕ) and the interval (0,1) by representing each subset as an infinite binary sequence, which corresponds to a binary decimal expansion. This connection reveals the deep relationship between set theory and analysis. 4. All Intervals Have the Same Cardinality

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