View source for Gimel function

In [[axiomatic set theory]], the '''gimel function''' is the following function mapping [[cardinal number]]s to cardinal numbers:

:<math>\gimel\colon\kappa\mapsto\kappa^{\mathrm{cf}(\kappa)}</math>

where cf denotes the [[cofinality]] function; the gimel function is used for studying the [[continuum function]] and the [[cardinal number#Cardinal exponentiation|cardinal exponentiation]] function. The symbol <math>\gimel</math> is a serif form of the Hebrew letter [[gimel]].

The '''gimel hypothesis''' states that <math>\gimel(\kappa)=\max(2^{\text{cf}(\kappa)},\kappa^+)</math>

==Values of the Gimel function==

The gimel function has the property  <math>\gimel(\kappa)>\kappa</math> for all infinite cardinals &kappa; by [[König's theorem (set theory)|König's theorem]].

For regular cardinals 
<math>\kappa</math>, 
<math>\gimel(\kappa)= 2^\kappa</math>, and [[Easton's theorem]] says we don't know much about the values of this function.   For singular 
<math>\kappa</math>, upper bounds for <math>\gimel(\kappa)</math> can be found from   [[Saharon Shelah|Shelah]]'s [[PCF theory]].

==Reducing the exponentiation function to the gimel function==

{{harvtxt|Bukovský|1965}} showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows. 

*If &kappa; is an infinite regular cardinal (in particular any infinite successor) then <math>2^\kappa = \gimel(\kappa)</math>
*If &kappa; is infinite and singular and the continuum function is eventually constant below &kappa; then <math>2^\kappa=2^{<\kappa}</math>
*If &kappa; is a limit and the continuum function is not eventually constant below &kappa; then <math>2^\kappa=\gimel(2^{<\kappa})</math>

The remaining rules hold whenever &kappa; and &lambda; are both infinite:

*If &alefsym;<sub>0</sub> &le; &kappa; &le; &lambda; then &kappa;<sup>&lambda;</sup> = 2<sup>&lambda;</sup>
*If &mu;<sup>&lambda;</sup> &ge; &kappa; for some &mu; < &kappa; then &kappa;<sup>&lambda;</sup> = &mu;<sup>&lambda;</sup>
*If &kappa; > &lambda; and &mu;<sup>&lambda;</sup> < &kappa; for all &mu; < &kappa; and cf(&kappa;) &le; &lambda; then &kappa;<sup>&lambda;</sup> = &kappa;<sup>cf(&kappa;)</sup>
*If &kappa; > &lambda; and &mu;<sup>&lambda;</sup> < &kappa; for all &mu; < &kappa; and cf(&kappa;) > &lambda; then &kappa;<sup>&lambda;</sup> = &kappa;

==References==

*{{citation|mr=0183649
|last=Bukovský|first= L.
|title=The continuum problem and powers of alephs 
|journal=Comment. Math. Univ. Carolinae |volume=6 |year=1965|pages= 181–197|url=http://hdl.handle.net/10338.dmlcz/105009}}
*{{citation|mr=0389593 
|last=Jech|first= Thomas J.
|title=Properties of the gimel function and a classification of singular cardinals 
|series=Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I. 
|journal=Fund. Math.|volume= 81 |year=1973|issue= 1|pages= 57–64|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm81/fm8116.pdf|doi=10.4064/fm-81-1-57-64}} 
*[[Thomas Jech]],  ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, {{ISBN|3-540-44085-2}}.

[[Category:Cardinal numbers]]