From what I understand:

  • The Wholeness Axiom(s) is/are the "ultimate axioms of infinity", bordering on inconsistency with ZFC.

  • Ultimate L (Completion of ZFC) attempts to extend the orderly world of constructible sets to encompass all large cardinals.

My question is: What happens when the two ultimates meet? Are they consistent ? If so, are the Wholeness Axioms the strongest large cardinal axioms consistent with Ultimate L ?

Edit: In response to Joel's comment below, I will modify my question to "What are the strongest large cardinal axioms know to be consistent with (or at least not inconsistent with) Ultimate L ?"

For instance, I believe Ultimate L => V = HOD. Are the rank-into-rank axioms consistent with this ? What about the WA's ? Anything else in the WA or rank-into-rank domain that may contradict Ultimate-L ?

  • 2
    $\begingroup$ I'm not sure how it affects your question, but there are several well-studied large cardinal axioms that are stronger than the wholeness axiom, such as rank-to-rank cardinals and others. See the upper attic in Cantor's attic: cantorsattic.info/Upper_attic. In consistency strength, the wholeness axiom is a weakening of $j:V_\lambda\to V_\lambda$, since $\langle V_\lambda,\in,j\rangle$ is a model of WA. $\endgroup$ – Joel David Hamkins Sep 23 '15 at 10:55
  • $\begingroup$ Thanks Joel. Cantor's Attic is a great resource ! I modified my question in response to your comment. $\endgroup$ – Cosmonut Sep 23 '15 at 19:43

I don't know to what extend the following answers your question, but a surprising result of Woodin says that the extension of the Inner Model Program to the level of one supercompact cardinal must yield the ultimate inner model, and then any such inner model necessarily inherits essentially all large cardinals from the universe of sets.

In his first paper ``Suitable extender models I'', Woodin shows that such an inner model would already relativize all the large cardinals up to having an elementary $j:V_λ\to V_λ$, which is much stronger than supercompactness itself.

In his second paper ``Suitable extender models II'' , Woodin proves even more and shows that many consistency-wise stronger hypotheses up to having a $j:L(V_{λ+1}) \to L(V_{λ+1})$ relativize to these models (Theorem 178).


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