|WikiProject Mathematics||(Rated Stub-class, Low-priority)|
Is λ or κ the rank-into-rank cardinal?
Regarding the current page contents, would it be safe to say that a cardinal λ is (a) rank-into-rank iff it satisfies one of the four axioms? (which?) -- Schnee 11:54, 18 Dec 2003 (UTC)
- Since I think that large cardinal properties which involve elementary embeddings belong to their critical points, I would call κ a rank-into-rank cardinal iff it is the critical point of any of the elementary embeddings mentioned in the definitions of I3, I2, I1, or I0. λ is larger than κ, but it is not as strong a limit, in fact, it has cofinality ω. JRSpriggs 05:58, 6 May 2006 (UTC)
Elementary Embedding Of Vλ For Non-Inaccessible λ?
The article mentions elementary embeddings of Vλ but also says that λ cannot be inaccessible (assuming choice). But an elementary embedding is an isomorphism between models, so if λ isn't inaccessible, what exactly is Vλ being considered as a model of? -- 220.127.116.11 (talk · contribs) 10:39, 6 August 2007 (UTC)
- A model of the theory of itself. Models are not required to be models "of" something given in advance. JRSpriggs (talk) 21:37, 24 January 2010 (UTC)
- Perhaps my previous answer was not sufficiently responsive. Vλ satisfies ZFC except for instances of the axiom of replacement where the image would have rank λ. In I2, M like V is a model of ZFC. Vλ+1 satisfies ZFC except for instances of replacement, pairing, or powerset where one of the given sets has rank λ. In I0, L(Vλ+1) satisfies ZF (without choice). I hope that helps. JRSpriggs (talk) 09:21, 29 January 2010 (UTC)
Is I0 inconsistent?
According to the article on Kunen's inconsistency theorem, one of its consequences says "If j is an elementary embedding of the universe V into an inner model M, and λ is the smallest fixed point of j above the critical point κ of j, then M does not contain the set j "λ (the image of j restricted to λ).". However according to this article, rank-into-rank axiom I0 says "There is a nontrivial elementary embedding of L(Vλ+1) into itself with the critical point below λ.". Now, j "λ is a subset of λ and thus an element of Vλ+1. If we apply Kunen's result to the submodel V' = M' = L(Vλ+1), does this not result in a contradiction since M' contains the forbidden element? JRSpriggs (talk) 21:37, 24 January 2010 (UTC)