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In `F33f`_`[axiomatic set theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Axiomatic_set_theory]`_`f, the `!gimel function`! is the following function mapping `F33f`_`[cardinal numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cardinal_number]`_`f to cardinal numbers:

ℷ ℷ : : κ κ ↦ ↦ κ κ c f ( κ κ ) {\\displaystyle \\gimel \\colon \\kappa \\mapsto \\kappa ^{\\mathrm {cf} (\\kappa )}}

where cf denotes the `F33f`_`[cofinality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cofinality]`_`f function; the gimel function is used for studying the `F33f`_`[continuum function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Continuum_function]`_`f and the `F33f`_`[cardinal exponentiation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cardinal_number]`_`f function. The symbol ℷ ℷ {\\displaystyle \\gimel } is a serif form of the Hebrew letter `F33f`_`[gimel`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gimel]`_`f.

>>Contents

• `F0af`_`[Values of the gimel function`#values-of-the-gimel-function]`_`f

• `F0af`_`[The gimel hypothesis`#the-gimel-hypothesis]`_`f

• `F0af`_`[Reducing the exponentiation function to the gimel function`#reducing-the-exponentiation-function-to-the-gimel-function]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Values of the gimel function

The gimel function has the property ℷ ℷ ( κ κ ) > κ κ {\\displaystyle \\gimel (\\kappa )>\\kappa } for all infinite cardinals κ κ {\\displaystyle \\kappa } by `F33f`_`[König's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=König's_theorem_(set_theory)]`_`f.

For regular cardinals κ κ {\\displaystyle \\kappa } , ℷ ℷ ( κ κ ) = 2 κ κ {\\displaystyle \\gimel (\\kappa )=2^{\\kappa }} , and `F33f`_`[Easton's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Easton's_theorem]`_`f says we don't know much about the values of this function. For singular κ κ {\\displaystyle \\kappa } , upper bounds for ℷ ℷ ( κ κ ) {\\displaystyle \\gimel (\\kappa )} can be found from `F33f`_`[Shelah`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Saharon_Shelah]`_`f's `F33f`_`[PCF theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PCF_theory]`_`f.

>>The gimel hypothesis

The `!gimel hypothesis`! states that ℷ ℷ ( κ κ ) = max ( 2 cf ( κ κ ) , κ κ + ) {\\displaystyle \\gimel (\\kappa )=\\max(2^{{\\text{cf}}(\\kappa )},\\kappa ^{+})} . In essence, this means that ℷ ℷ ( κ κ ) {\\displaystyle \\gimel (\\kappa )} for singular κ κ {\\displaystyle \\kappa } is the smallest value allowed by the axioms of `F33f`_`[Zermelo–Fraenkel set theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zermelo–Fraenkel_set_theory]`_`f (assuming consistency).

Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the `F33f`_`[continuum hypothesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Continuum_hypothesis]`_`f (which implies the gimel hypothesis).

>>Reducing the exponentiation function to the gimel function

`F33f`_`[Bukovský (1965)`#citerefbukovsk-1965]`_`f showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows.

• If κ κ {\\displaystyle \\kappa } is an infinite regular cardinal (in particular any infinite successor) then 2 κ κ = ℷ ℷ ( κ κ ) {\\displaystyle 2^{\\kappa }=\\gimel (\\kappa )}

• If κ κ {\\displaystyle \\kappa } is infinite and singular and the continuum function is eventually constant below κ κ {\\displaystyle \\kappa } then 2 κ κ = 2 < κ κ {\\displaystyle 2^{\\kappa }=2^{<\\kappa }}

• If κ κ {\\displaystyle \\kappa } is a limit and the continuum function is not eventually constant below κ κ {\\displaystyle \\kappa } then 2 κ κ = ℷ ℷ ( 2 < κ κ ) {\\displaystyle 2^{\\kappa }=\\gimel (2^{<\\kappa })}

The remaining rules hold whenever κ κ {\\displaystyle \\kappa } and λ λ {\\displaystyle \\lambda } are both infinite:

• If ℵ0 ≤ `*κ`* ≤ `*λ`* then `*κλ`* = 2`*λ`*

• 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`* = `*κ`*cf(κ)

• If `*κ`* > `*λ`* and `*μλ`* < `*κ`* for all `*μ`* < `*κ`* and cf(`*κ`*) > `*λ`* then `*κλ`* = `*κ`*

>>See also

• `F33f`_`[Aleph number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Aleph_number]`_`f

• `F33f`_`[Beth number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Beth_number]`_`f

>>References

• `:citerefbukovsk-1965`aBukovský, L. (1965), "The continuum problem and powers of alephs", `*Comment. Math. Univ. Carolinae`*, `!6`!: 181–197, `F33f`_`[hdl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hdl_(identifier)]`_`f:10338.dmlcz/105009, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0183649

• `:citerefjech1973`aJech, Thomas J. (1973), "Properties of the gimel function and a classification of singular cardinals" (PDF), `*Fund. Math.`*, Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I., `!81`! (1): 57–64, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.4064/fm-81-1-57-64, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0389593

• `F33f`_`[Thomas Jech`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thomas_Jech]`_`f, `*Set Theory`*, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-540-44085-2.

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