Ordinal Arithmetic

Set Theory · Axiom Academy

Addition, multiplication, and exponentiation on ordinals: order matters! Ordinal addition α + β is defined by concatenation : place a copy of β after a copy of α. 2 + 3 = 5 (as expected from natural numbers) 1 + ω = ω (one element followed by ω elements = ω) ω + 1 ≠ ω (ω elements followed by one element ≠ ω) 2. Non-Commutativity of Addition Unlike natural number addition, ordinal addition is not commutative : α + β ≠ β + α in general. Concatenate ω = 0, 1, 2, 3, ... after it Result: 0 followed by 0, 1, 2, 3, ... This is order-isomorphic to ω itself (the first 0 is "absorbed") Start with ω = 0, 1, 2, 3, ... Add one element after all of them Result: 0, 1, 2, 3, ..., ω which has a largest element This is fundamentally different from ω (a limit ordinal with no largest element) Ordinal multiplication α · β means "β copies of α, concatenated in order." 2 · 3 = 6 (three copies of 2 = 0,1 + 0,1 + 0,1 ) ω · 2 = ω + ω (two copies of ω) 2 · ω = ω (ω copies of 2, but this equals ω!) Ordinal exponentiation α^β represents iterated multiplication. ω^ω = sup ω^0, ω^1, ω^2, ω^3, ... 2^ω = sup 2^0, 2^1, 2^2, ... = sup 1, 2, 4, 8, ... = ω The last example shows another non-commutativity: 2^ω = ω, but ω^2 = ω·ω is much larger! Let's consolidate our understanding with important examples. 1 + ω = ω (adding before a limit doesn't change it) ω + 1 = ω + 1 (adding after creates a new ordinal) 2 · ω = ω (finite times infinity = infinity) ω · 2 = ω + ω (infinity times finite > infinity)

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