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We call a set $X ⊆ V_{λ+1}$ an Icarus set if there is an elementary embedding $j : L(X, V_{λ+1}) ≺ L(X, V_{λ+1})$ with $\mathrm{crit}(j)< λ$.

But this raises the question: What is the most "Icarus" Icarus set like?

If $V_{λ+1}$ is Icarus, is strongly implied by $(V_{λ+1})^{♯}$, that is strongly implied by $(V_{λ+1})^{♯♯}$. [Dimonte, 2017]

If $\Theta^{M}<\Theta^{L(X, V_{λ+1})}$ then we consider “X is an Icarus set” as something stronger than “M is an Icarus set”. [Dimonte, 2011]

Even with "$M=(V_{λ+1})^{<\gamma♯}$" or "$M=E^0_{\infty}(V_{λ+1})$" or "$M=E^{j_{<\omega}}_{\infty}(V_{λ+1})$", according to the previous deduction, they cannot be the most "Icarus" Icarus set.

I looked up the literature [Woodin, 2011]. AD-Conjecture is indeed very good, but it doesn't seem to solve the problem. Of course I can't make sure I'm not missing anything, it's too long.


2022-04-18:

Well, I now probably understand that AD-Conjecture can hold an Icarus set with the maximum consistency strength. But this does not convince me that:

  1. Perhaps there could also be a forcing-axiom-style (Martin's maximum? Ω-logic?)/powerset-maximal-style (V-logic?) Icarus set with maximum consistency strength based on their set-theoretic philosophy.
  2. There is no known relative consistency analysis/ordinal analysis conclusion that the candidates are indeed the ones with the maximum consistency strength.
  3. Does it really require a very large consistency strength just to wish to hold all sets $X ⊆ V_{λ+1}$ that can be used for embedding? What is the difference between this and Kunen inconsistency that prevents us from proving it in ZFC?

References

[Dimonte, 2011] Dimonte, V. (2011). Very Large Cardinals and Elementary Embeddings Properties

[Dimonte, 2017] Dimonte, V. (2017). I0 and rank-into-rank axioms.

[Woodin, 2011] Woodin, W. H. (2011). Suitable extender models II: beyond ω-huge.

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  • $\begingroup$ How would the following quote from Dimonte's "$I0$ and rank-into-rank axioms" (pg. 61) help you answer your question (if at all): "Trivially, if $I0(\lambda)$ holds, (and in fact every set in $L(V_{\lambda + 1})$ $\cap$ $V_{\lambda + 2}$ ) is Icarus. On the other hand, a well-ordering of $V_{\lambda + 1} $cannot be Icarus, because of Kunen's theorem." $\endgroup$ May 16, 2022 at 5:59
  • $\begingroup$ *$V_{\lambda + 1}$ (and in fact... $\endgroup$ May 16, 2022 at 6:20
  • $\begingroup$ @ThomasBenjamin 1. $E^0_0=L(V_{\lambda + 1})\cup V_{\lambda + 2}$, So $E^0_1$ is more "Icarus" than it. 2. $I0(\lambda)$ is itself a dangerous and alleged violation of the axioms of Kunen inconsistency and HOD conjecture. Because $Con(ZF + \lambda\text{-}DC + I0(\lambda))\to Con(ZF + \lambda\text{-}DC + I0(\lambda) + j\uparrow V_{\lambda+2})$. [*]Schlutzenberg, F. (2020). On the consistency of ZF with an elementary embedding from $ V_ {\lambda+ 2} $ into $ V_ {\lambda+ 2} $. arxiv.org/abs/2006.01077 $\endgroup$ May 17, 2022 at 19:35
  • $\begingroup$ Thanks for the reference. I'll take a look. $\endgroup$ May 18, 2022 at 1:42
  • $\begingroup$ In '1', shouldn't the $\cup$ be a $\cap$ in the definition of $E^{0}_{0}$? $\endgroup$ May 18, 2022 at 17:26

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