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A cardinal number measures the size of a set via bijection: two sets have the same cardinality if and only if there is a bijection between them. In ZFC, each infinite cardinal is represented as an initial ordinal — an ordinal not in bijection with any smaller ordinal. The infinite cardinals are indexed by ordinals via the aleph sequence: ℵ₀ (cardinality of ℕ), ℵ₁ (the next uncountable cardinal), ℵ₂, ..., ℵ_ω, .... The axiom of choice ensures every set has a cardinality comparable with all others; without choice, some sets cannot be well-ordered and thus have no aleph. The aleph hierarchy, defined by transfinite recursion on ordinals, provides a complete listing of all infinite cardinals.
Begin with countability and the distinction between ℵ₀ and uncountable sets. Then define initial ordinals formally: verify ω is the initial ordinal for ℵ₀, and construct ω₁ (the first uncountable ordinal) via the Hartogs number construction. Build the first few alephs and locate familiar sets (ℕ, ℚ, ℝ) within the hierarchy.
From your study of cardinality and countability, you know that two sets have the same size if and only if there is a bijection between them, and that infinite sets can have different cardinalities — Cantor's diagonal argument shows ℕ and ℝ are not in bijection. Infinite cardinal numbers are the formal system for measuring and comparing the sizes of infinite sets within ZFC.
In ZFC, each infinite cardinal is identified with a specific ordinal: an *initial ordinal*, meaning an ordinal that is not in bijection with any smaller ordinal. The smallest infinite cardinal is ℵ₀ = ω, the order-type of the natural numbers. The next infinite cardinal ℵ₁ is constructed as ω₁, the Hartogs number of ω: the set of all countable ordinals, which turns out to be the smallest uncountable ordinal. By the axiom of replacement, this set exists and has a well-defined cardinality strictly greater than ℵ₀. The process continues by transfinite recursion: ℵ_{α+1} is the Hartogs number of ℵ_α, and ℵ_λ = sup{ℵ_α : α < λ} for limit ordinals λ. This gives the aleph sequence ℵ₀, ℵ₁, ℵ₂, ..., ℵ_ω, ℵ_{ω+1}, ... indexing all infinite cardinals.
The relationship between ordinals and cardinals is important to keep straight. Ordinals measure order-type — ω and ω+1 are different ordinals because they have different well-order structures. But as sets they are in bijection (just move the last element), so they have the same cardinality ℵ₀. Cardinals are the initial ordinals: among all ordinals of a given cardinality, the smallest one. So ℵ₁ = ω₁, the smallest uncountable ordinal, is both the first uncountable cardinal and an ordinal. Every cardinal is an ordinal, but most ordinals are not cardinals.
A common and significant misconception: ℵ₁ is not defined as |ℝ|. The real numbers have cardinality 2^ℵ₀ (the size of the power set of ℕ), and Cantor's theorem guarantees 2^ℵ₀ > ℵ₀, so ℝ is uncountable. But 2^ℵ₀ could be ℵ₁, or ℵ₂, or ℵ_ω, or many other cardinals. Whether 2^ℵ₀ = ℵ₁ is exactly the continuum hypothesis (CH). Gödel showed in 1940 that ZFC cannot disprove CH, and Cohen showed in 1963 that ZFC cannot prove CH. CH is independent of ZFC — it is a genuinely undecidable statement in standard set theory.
The axiom of choice plays a hidden but essential role throughout this theory. Without the axiom of choice, not every set can be well-ordered, and therefore not every set has a cardinality in the aleph hierarchy. The axiom of choice guarantees that every set is in bijection with some initial ordinal, making the aleph sequence a complete classification of all cardinalities. This is why the axiom of choice is not merely a convenience but a structural prerequisite for the theory of infinite cardinals to work as described.