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In [[set theory]], '''Berkeley cardinals''' are certain [[large cardinal]]s suggested by [[Hugh Woodin]] in a seminar at the [[University o ...rkeley|access-date=2023-04-15|website=neugierde.github.io}}</ref> Berkeley cardinals are a strictly stronger cardinal axiom than [[Reinhardt cardinal]]s, implyi ...

...en's inconsistency theorem]] uses a Jónsson function for [[cardinal number|cardinals]] λ such that 2^λ = λ^ℵ₀, and Kunen observed ...tion | last1=Kanamori | first1=Akihiro | title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|title-link=The Higher Infinite | publis ...

...)|ultrafilters]] which represents an [[elementary embedding]] witnessing [[large cardinal]] properties. A nonprincipal ultrafilter is the most basic case of Let &kappa; and &lambda; be cardinals with &kappa;&le;&lambda;. Then, a set <math>E = \{E_a | a\in [\lambda]^{<\o ...

...tive]], or even [[well-founded]], provided ''κ'' has sufficiently strong [[large cardinal]] properties. Well-foundedness fails specifically for [[rank-into- ...s-Mitchell/ecf7380a4468e233a23282157b318e20156e3a1a Inner models for large cardinals] (2012, p.8). Accessed 2022-12-07.</ref> ...

...r = [[Mathematical logic]], [[Axiomatic Set Theory|Set theory]], [[Large cardinal property]] ...rom the [[Hebrew University of Jerusalem]]. His thesis, ''On Super Compact Cardinals'', was written under the supervision of [[Azriel Lévy]].<ref>{{MathGenealog ...

...rlink=Kenneth Kunen|last=Kunen|year=1971}}, shows that several plausible [[large cardinal]] axioms are [[Consistency|inconsistent]] with the [[axiom of choi ...e elementary embeddings into themselves, at least if one assumes some mild large cardinal axioms. For example, if [[zero sharp|0^#]] exists then t ...

...s-Mitchell/ecf7380a4468e233a23282157b318e20156e3a1a Inner models for large cardinals] (2012, p.16). Accessed 2022-12-08.</ref> and finally {{harvs|txt|authorli ...heorem is also true: if 0^# exists then the countable set of all cardinals less than <math>\aleph_\omega</math> cannot be covered by a constructible s ...

...ses [[Mahlo cardinal|weakly Mahlo cardinals]] <math>M</math> to generate [[large countable ordinal]]s.<ref>{{Cite web|last=Rathjen|first=Michael|date=1990|t ...st=Michael|date=1994-01-01|title=Collapsing functions based on recursively large ordinals: A well-ordering proof for KPM|url=https://doi.org/10.1007/BF01275 ...

...lecture was given by Ronald Jensen, who spoke on ''Inner Models and Large Cardinals.''}}</ref><ref>{{Cite journal|date=1992|title=Annual Meeting of the Associa * 1990 [[Ronald Jensen]], ''Inner Models and Large Cardinals.'' ...

While (without the existence of [[large cardinal]]s) there are examples of non-tame AECs,<ref>{{harvnb|Baldwin|Shel * '''Tameness is a large cardinal axiom''':<ref>{{harvnb|Boney|2014}}, Theorem 1.3.</ref> There are ...

...[[set theory|set-theoretic]] assumptions (such as the existence of [[large cardinals]] or variations of the [[generalized continuum hypothesis]]), or model-theo ...math>\beth_{(2^{\operatorname{LS}(K)})^+}</math> has models of arbitrarily large sizes. ...

...r1=Joel David Hamkins |author2=Hans Robin Solberg |title=Categorical large cardinals and the tension between categoricity and set-theoretic reflection |date=202 ...

...x(yz)=(xy)(xz),</math> a property that is a priori unconnected with large cardinals.<ref>{{Citation | last1= Dehornoy | first1= Patrick | title= Sur la structu ....<ref name = Dehornoy94/> Since the braid order appears precisely when the large cardinal assumption is eliminated, the link between the braid order and the ...

...t, however it is usually clear from context.<ref>K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991 ...nd exponentiation <ref>F. R. Drake, ''Set Theory: An Introduction to Large Cardinals'' (p.83). Accessed 1 July 2022.</ref> ...

...r = [[Mathematical logic]], [[Axiomatic Set Theory|Set theory]], [[Large cardinal property]] *{{cite journal|title=Genericity and large cardinals|author= Friedman, Sy D|journal=J. Math. Log. |year=2005|volume=5|issue=2|pa ...

* Akihiro Kanamori: ''[[The Higher Infinite|The Higher Infinite. Large Cardinals in Set Theory from their Beginnings]].'', Perspectives in Mathematical Logi ...

...b>n</sub>(''x_i'')=&lambda;_''n'' for all sufficiently large ''i'', then ...] |editor3-first=John R.|editor3-last=Steel|title=Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I |publisher=Cambridge University Press |year=20 ...

...ch measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other jus ...ifficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual [[Zermelo–Fraenkel set theory|Zermelo–Fraenkel]] ( ...

{{defn|no=1|defn=A product of cardinals}} {{defn|no=1|defn=A sum of cardinals}} ...

...ability of S2S is the best possible. Graphs with unbounded treewidth have large grid minors, which can be used to simulate a [[Turing machine]]. ...cond order logic on ordinals iff it can be obtained from definable regular cardinals by ordinal addition and multiplication.<ref>{{Citation |chapter=Monadic def ...