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In the von Neumann representation, each ordinal is defined as the set of all smaller ordinals: 0 = ∅, 1 = {0}, 2 = {0,1}, ω = {0,1,2,...}, ω+1 = {0,1,2,...,ω}, and so on. A set α is a (von Neumann) ordinal if it is transitive (every element of α is also a subset of α) and is well-ordered by membership ∈. Every well-ordered set is order-isomorphic to a unique ordinal, making ordinals canonical representatives of well-order types. The ordinals are partitioned into three kinds: 0 (the empty set), successor ordinals (of the form α ∪ {α}), and limit ordinals (non-zero ordinals with no immediate predecessor, like ω, ω·2, ε₀).
Build the first several ordinals explicitly: 0, 1, 2, 3, ω, ω+1, ω+2, ω+ω. For each, verify transitivity and that ∈ well-orders the set. Work through the proof that any well-ordered set is isomorphic to a unique ordinal — this makes the definition feel canonical rather than arbitrary.
The von Neumann construction answers a deceptively simple question: if we want to represent the natural numbers (and beyond) purely as sets, what should each number *be*? The answer is elegant — each ordinal is the set of all smaller ordinals. So 0 = ∅ (no smaller ordinals exist), 1 = {0} = {∅}, 2 = {0, 1} = {∅, {∅}}, 3 = {0, 1, 2}, and so on. The pattern means the ordinal n always has exactly n elements, and the membership relation ∈ on any finite ordinal behaves exactly like the "less than" relation on natural numbers.
To move beyond the finite, the axiom of infinity guarantees the existence of a set containing all finite ordinals. That set is ω = {0, 1, 2, 3, ...} — the first infinite ordinal. ω is not just a symbol for "infinity"; it is a specific, well-defined set. After ω come ω+1 = ω ∪ {ω}, ω+2, and eventually ω+ω (written ω·2), then ω·3, ω², and so on. These transfinite ordinals form a proper class — there is no set of all ordinals.
Ordinals come in three kinds. The ordinal 0 = ∅ is the base case. A *successor ordinal* is one of the form α ∪ {α}, written α+1; every finite ordinal and ω+1, ω+2, ... are successors. A *limit ordinal* is a non-zero ordinal with no immediate predecessor; ω is the first, and ω·2, ω², ε₀ are further examples. This trichotomy is fundamental to transfinite induction and recursion.
The formal definition requires two properties. An ordinal α must be *transitive*: every element of α is also a subset of α. This ensures α is "closed downward" and contains all the structure of its predecessors. It must also be *well-ordered by ∈*: every non-empty subset has a least element. Together, these conditions force a unique canonical form — there is exactly one von Neumann ordinal for each order type of a well-ordered set.
One important warning: ordinal arithmetic is *not* commutative. Adding 1 before ω gives ω (still a countable sequence), but adding 1 after ω gives ω+1 (which has a new top element). This asymmetry reflects the fact that ordinals encode *ordered* structure, not just size. When you later encounter cardinal numbers, you will find a different arithmetic that *is* commutative for infinite cardinals — the contrast with ordinal arithmetic is instructive.