KLEENE'S 𝒪
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1
Successor
Notation: 1 = 2^0
The first successor ordinal: 0 + 1. In Kleene's O, written as 2^0.
CONSTRUCTION
If a is in O with |a| = α, then 2^a is in O with |2^a| = α + 1.
SIGNIFICANCE
The successor operation is the simplest way to build new ordinals — each step adds one.
Recursive ordinals · ω₁ᶜᶾ boundary · rotates daily