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fixed points of normal functions (Definition)

If $ f\colon M\rightarrow\mathbf{On}$ is a function from any set of ordinals to the class of ordinals then $ \operatorname{Fix}(f)=\{x\in M\mid f(x)=x\}$ is the set of fixed points of $ f$. $ f^\prime$, the derivative of $ f$, is the enumerating function of $ \operatorname{Fix}(f)$.

If $ f$ is $ \kappa$-normal then $ \operatorname{Fix}(f)$ is $ \kappa$-closed and $ \kappa$-normal, and therefore $ f^\prime$ is also $ \kappa$-normal.

For example, the function which takes an ordinal $ \alpha$ to the ordinal $ 1+\alpha$ has a fixed point at every ordinal $ \geq\omega$, so $ f^\prime(\alpha)=\omega+\alpha$.



"fixed points of normal functions" is owned by Henry.
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See Also: proof of the power rule, Leibniz notation, proof of product rule, proof of sum rule, sum rule, directional derivative, Newton's method

Also defines:  derivative

Attachments:
proof of fixed points of normal functions (Proof) by Henry
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Cross-references: enumerating function, fixed points, class of ordinals, ordinals, function
There are 67 references to this entry.

This is version 3 of fixed points of normal functions, born on 2003-02-23, modified 2004-10-10.
Object id is 4054, canonical name is FixedPointsOfNormalFunctions.
Accessed 10267 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
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