Normal function

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In axiomatic set theory, the function f : Ord -> Ord is called normal (or normal function ) if continuous (with respect to the order topology) and monotonously increased strictly. This is equivalent to the following two conditions:

1. For each ordinal limit? (ie? not zero or substitute), f (?) = soup { f (?) Ã,:? & lt; ?}.
2. For all ordinal? & lt; ?, f (?) & lt; f (?).

Video Normal function

Example

A simple normal function is given by f (?) = 1? (see ordinal arithmetic). But f (?) =? 1 is not normal. If is ordinal fixed, then function f (?) =? ?, f (?) =? A-? (for?> = 1), and f (?) =? ^? (for?> = 2) everything is normal.

Contoh yang lebih penting dari fungsi normal diberikan oleh nomor aleph ${\ displaystyle f (\ alpha) = \ aleph_ {\ alpha}} Â Â$ yang menghubungkan nomor ordinal dan kardinal, dan dengan nomor beth ${\ displaystyle f (\ alpha) = \ beth _ {\ alpha}} Â Â$ .

Maps Normal function

Properti

If f is normal, then for any ordinal ?,

f (?)> =?.

Evidence : Otherwise, choose? at least like that f (?) & lt; ? Because f is monotonously increased, f ( f (?)) & Lt; f (?), contrary to the minimality?

Furthermore, for any non-empty ordinal set, we have

f (sup S ) = so f ( S ).

Evidence : "> =" follows from the monotony of f and the supremum definition. For "<=", set it? = soup S and consider three cases:

• if? = 0, then S = {0} and soup f ( S ) = f (0);
• if? =? 1 is the successor, then there is s in S with? & lt; s , so is it? <= s . Therefore, f (?) <= f ( s ), which implies f (?) < = Sup f ( S );
• if? is a non-zero limit, choose? & lt; ? and s in S like that? & lt; s (probably because? = soup S ). Therefore, f (?) & Lt; f ( s ) so f (?) & lt; soup f ( S ), resulting in f (?) = soup { f (?) Ã,:? & lt; ?} <= sup f ( S ), as desired.

Each normal function f has an arbitrarily large fixed point; see fixed point lemma for normal function as evidence. One can create the normal function of f ': Ord -> Ord, called derivative of f , so f' (?) Is the fixed point-to-1 f .

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Note

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References

Johnstone, Peter (1987), Note on Logic and Set Theory , Cambridge University Press, ISBN 978-0-521 -33692-5 .

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