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:

- 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.
- There is no known relative consistency analysis/ordinal analysis conclusion that the candidates are indeed the ones with the maximum consistency strength.
**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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