Derivative (set theory)

In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is 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. 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 functions.

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