Extendible cardinal

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In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivated by reflection principles. A cardinal number κ is called η-extendible if for some λ there is a nontrivial elementary embedding j of




where κ is the critical point of j.

κ is called an extendible cardinal if it is η-extendible for every ordinal number η.

Vopěnka's principle implies the existence of extendible cardinals. All extendible cardinals are supercompact cardinals.

See also[edit]


"A cardinal κ is extendible if and only if for all α>κ there exists β and an elementary embedding from V(α) into V(β) with critical point κ." -- "Restrictions and Extensions" by Harvey M. Friedman http://www.math.ohio-state.edu/~friedman/pdf/ResExt021703.pdf

  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3. 
  • Reinhardt, W. N. (1974), "Remarks on reflection principles, large cardinals, and elementary embeddings.", Axiomatic set theory, Proc. Sympos. Pure Math., XIII, Part II, Providence, R. I.: Amer. Math. Soc., pp. 189–205, MR 0401475