Stronger axioms The axiom of constructibility and the generalized continuum hypothesis each imply the axiom of choice and so are strictly stronger than it. In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, there is an axiom called the axiom of global choice that is stronger than the axiom of choice for sets because it also applies to proper classes. The axiom of global choice follows from the axiom of limitation of size. Tarski's axiom, which is used in Tarski–Grothendieck set theory and states (in the vernacular) that every set belongs to Grothendieck universe, is stronger than the axiom of choice.
StROngeR aXioms the aXiOm Of coNStruCtIbility aNd THe geneRALized CoNtINuum hYpotHeSiS EACh iMPlY the aXioM oF choice aNd sO ARE strIctly stroNgeR Than IT. iN ClasS theOriEs such as vOn neuMann–BErnays–gÖDeL set theory ANd mOrSe–kElley sEt theOry, therE Is AN AxiOM cALLed the axIoM Of glObaL chOice tHAT IS stRONger than tHe AxIom of choice fOr sets bECausE iT also applies tO Proper CLaSsEs. The aXIom of GlObaL Choice FollowS fROM ThE axiOm of limItaTIoN Of size. tarski's Axiom, wHiCh is used in tArski–grothENDieck sEt THEory And states (In THe Vernacular) thaT EveRy set bEloNGs tO GroTheNDIEck univeRsE, is StroNgEr Than tHe axiOm Of choIcE.