Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Gimel function
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{Short description|Theorem in axiomatic set theory}} 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]]. ==Values of the gimel function== The gimel function has the property <math>\gimel(\kappa)>\kappa</math> for all infinite cardinals <math>\kappa</math> 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]]. ==The gimel hypothesis== The '''gimel hypothesis''' states that <math>\gimel(\kappa)=\max(2^{\text{cf}(\kappa)},\kappa^+)</math>. In essence, this means that <math>\gimel(\kappa)</math> for singular <math>\kappa</math> is the smallest value allowed by the axioms of [[Zermelo–Fraenkel set theory]] (assuming consistency). Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the [[continuum hypothesis]] (which implies the gimel hypothesis). ==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 <math>\kappa</math> is an infinite regular cardinal (in particular any infinite successor) then <math>2^\kappa = \gimel(\kappa)</math> *If <math>\kappa</math> is infinite and singular and the continuum function is eventually constant below <math>\kappa</math> then <math>2^\kappa=2^{<\kappa}</math> *If <math>\kappa</math> is a limit and the continuum function is not eventually constant below <math>\kappa</math> then <math>2^\kappa=\gimel(2^{<\kappa})</math> The remaining rules hold whenever <math>\kappa</math> and <math>\lambda</math> are both infinite: *If {{math|ℵ<sub>0</sub> ≤ {{var|κ}} ≤ {{var|λ}}}} then {{math|1=''κ<sup>λ</sup>'' = 2<sup>''λ''</sup>}} *If {{math|{{var|μ<sup>λ</sup>}} ≥ {{var|κ}}}} for some {{math|{{var|μ}} < {{var|κ}}}} then {{math|1=''κ<sup>λ</sup>'' = ''μ<sup>λ</sup>''}} *If {{math|{{var|κ}} > {{var|λ}}}} and {{math|{{var|μ<sup>λ</sup>}} < {{var|κ}}}} for all {{math|{{var|μ}} < {{var|κ}}}} and {{math|cf(''κ'') ≤ ''λ''}} then {{math|1=''κ<sup>λ</sup>'' = ''κ''<sup>cf(κ)</sup>}} *If {{math|{{var|κ}} > {{var|λ}}}} and {{math|{{var|μ<sup>λ</sup>}} < {{var|κ}}}} for all {{math|{{var|μ}} < {{var|κ}}}} and {{math|cf(''κ'') > ''λ''}} then {{math|1=''κ<sup>λ</sup>'' = ''κ''}} ==See also== * [[Aleph number]] * [[Beth number]] ==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|hdl=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|doi-access=free}} *[[Thomas Jech]], ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, {{ISBN|3-540-44085-2}}. [[Category:Cardinal numbers]]
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)
Pages transcluded onto the current version of this page
(
help
)
:
Template:Citation
(
edit
)
Template:Harvtxt
(
edit
)
Template:ISBN
(
edit
)
Template:Math
(
edit
)
Template:Short description
(
edit
)