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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.
 
==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==
 
All cardinal exponentiation is determined (recursively) by the gimel function as follows.  
*If &kappa; is an infinite  successor cardinal then <math>2^\kappa = \gimel(\kappa)</math>
*If &kappa; is a limit and the continuum function is eventually constant below &kappa; then <math>2^\kappa=2^{<\kappa}\times\gimel(\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==
*[[Thomas Jech]],  ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.
 
[[Category:Cardinal numbers]]

Revision as of 12:52, 12 October 2013

In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers:

where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal exponentiation function.

Values of the Gimel function

The gimel function has the property for all infinite cardinals κ by König's theorem.

For regular cardinals , , and Easton's theorem says we don't know much about the values of this function. For singular , upper bounds for can be found from Shelah's PCF theory.

Reducing the exponentiation function to the gimel function

All cardinal exponentiation is determined (recursively) by the gimel function as follows.

The remaining rules hold whenever κ and λ are both infinite:

  • If ℵ0≤κ≤λ then κλ = 2λ
  • If μλ≥κ for some μ<κ then κλ = μλ
  • If κ> λ and μλ<κ for all μ<κ and cf(κ)≤λ then κλ = κcf(κ)
  • If κ> λ and μλ<κ for all μ<κ and cf(κ)>λ then κλ = κ

References

  • Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.