Derivative (set theory)
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In
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
(with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions: # For every limit ordinal ''γ'' (i.e. ''γ'' is neither zero nor a successor), it is the case that ''f''(''γ'') = sup . # For all ordinals ''α'' < ''β'', it is the case that ''f''(''α'') < ''f''(''β'').


Examples

A simple normal function is given by (see ordinal arithmetic). But is ''not'' normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when ''λ'' is a limit ordinal. If ''β'' is a fixed ordinal, then the functions , (for ), and (for ) are all normal. More important examples of normal functions are given by the aleph numbers f(\alpha) = \aleph_\alpha, which connect ordinal and cardinal numbers, and by the beth numbers f(\alpha) = \beth_\alpha.


Properties

If ''f'' is normal, then for any ordinal ''α'', :''f''(''α'') ≥ ''α''. Proof: If not, choose ''γ'' minimal such that ''f''(''γ'') < ''γ''. Since ''f'' is strictly monotonically increasing, ''f''(''f''(''γ'')) < ''f''(''γ''), contradicting minimality of ''γ''. Furthermore, for any non-empty set ''S'' of ordinals, we have :''f''(sup ''S'') = sup ''f''(''S''). Proof: "≥" follows from the monotonicity of ''f'' and the definition of the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
. For "≤", set ''δ'' = sup ''S'' and consider three cases: * if ''δ'' = 0, then ''S'' = and sup ''f''(''S'') = ''f''(0); * if ''δ'' = ''ν'' + 1 is a successor, then there exists ''s'' in ''S'' with ν < ''s'', so that ''δ'' ≤ ''s''. Therefore, ''f''(''δ'') ≤ ''f''(''s''), which implies ''f''(δ) ≤ sup ''f''(''S''); * if ''δ'' is a nonzero limit, pick any ''ν'' < ''δ'', and an ''s'' in ''S'' such that ν < ''s'' (possible since ''δ'' = sup ''S''). Therefore, ''f''(''ν'') < ''f''(''s'') so that ''f''(''ν'') < sup ''f''(''S''), yielding ''f''(''δ'') = sup ≤ sup ''f''(''S''), as desired. Every normal function ''f'' has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function ''f' '' : Ord → Ord, called the derivative of ''f'', such that ''f' ''(''α'') is the ''α''-th fixed point of ''f''. For a hierarchy of normal functions, see
Veblen function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...
s.


Notes


References

*{{citation , first=Peter , last=Johnstone , authorlink=Peter Johnstone (mathematician) , year=1987 , title=Notes on Logic and Set Theory , publisher= Cambridge University Press , isbn=978-0-521-33692-5 , url-access=registration , url=https://archive.org/details/notesonlogicsett0000john . Set theory Ordinal numbers