Rank-into-rank

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In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):

  • Axiom I3: There is a nontrivial elementary embedding of Vλ into itself.
  • Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point.
  • Axiom I1: There is a nontrivial elementary embedding of Vλ+1 into itself.
  • Axiom I0: There is a nontrivial elementary embedding of L(Vλ+1 ) into itself with the critical point below λ.

These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom for Reinhardt cardinals is stronger, but is not consistent with the axiom of choice.

If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit of jn(κ) as n goes to ω. More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementary embedding of Vα into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal.

The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen's result that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.

[edit] References

  • Gaifman, Haim (1974), “Elementary embeddings of models of set-theory and certain subtheories”, Axiomatic set theory, vol. XIII, Part II, Proc. Sympos. Pure Math., Providence R.I.: Amer. Math. Soc., pp. 33--101, MR0376347 
  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3-540-00384-3. 
  • Solovay, Robert M.; Reinhardt, William N. & Kanamori, Akihiro (1978), “Strong axioms of infinity and elementary embeddings”, Annals of Mathematical Logic 13 (1): 73–116, DOI doi:10.1016/0003-4843(78)90031-1 
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