Epsilon Numbers

Set Theory · Axiom Academy

Fixed points of ordinal exponentiation where ω ε = ε. 1. Fixed Points of Exponentiation An epsilon number is an ordinal ε such that ω^ε = ε. These are the fixed points of the function f(α) = ω^α. Epsilon numbers represent thresholds where exponential growth "catches up" with the ordinal itself. We can construct ε₀ as the supremum of an increasing sequence of ordinals: Each step applies ω^(-) to the previous result, and ε₀ is where this process stabilizes. ε₀ can be visualized as an infinite tower of ω's: This tower never collapses—each level perfectly supports the level above. ε₀ is just the first epsilon number. There are infinitely many epsilon numbers, forming their own hierarchy. Each epsilon number is a fixed point, and the collection of all epsilon numbers is itself unbounded in the ordinals. 5. Applications in Proof Theory Epsilon numbers appear in proof theory as measures of the strength of formal systems. Epsilon numbers mark important boundaries in the landscape of ordinals and logical strength.

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