$\exists j:L(V_{\lambda+1})\to L(V_{\lambda+1})$

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See first: rank into rank axioms

The axiom I0, the large cardinal axiom of the title, asserts that some nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ extends to a nontrivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$, where $L(V_{\lambda+1})$ is the transitive proper class obtained by starting with $V_{\lambda+1}$ and forming the constructible hierarchy over $V_{\lambda+1}$ in the usual fashion (see constructible universe). An alternate, but equivalent formulation asserts the existence of some nontrivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with $\mathrm{crit}(j) < \lambda$. The critical point assumption is essential for the large cardinal strength as otherwise the axiom would follow from the existence of some measurable cardinal above $\lambda$. The axiom is of rank into rank type, despite its formulation as an embedding between proper classes, and embeddings witnessing the axiom known as $\text{I0}$ embeddings.

Originally formulated by Woodin in order to establish the relative consistency of a strong determinacy hypothesis, it is now known to be obsolete for this purpose (it is far stronger than necessary). Nevertheless, research on the axiom and its variants is still widely pursued and there are numerous intriguing open questions regarding the axiom and its variants, see [1].

The axiom subsumes a hierarchy of the strongest large cardinals not known to be inconsistent with $\text{ZFC}$ and so is seen as "straining the limits of consistency" [1]. An immediate observation due to the Kunen inconsistency is that, under the $\text{I0}$ axiom, $L(V_{\lambda+1})$ cannot satisfy the axiom of choice.

The $\mathrm{I0}$ axiom is sometimes called $\tilde{L}$[2].

Ultrapower Reformulation

Despite the class language formulation of $I_0$, there is a first-order formulation in terms of normal ultrafilters: define, for $j:L(V_{\lambda + 1})\prec L(V_{\lambda+1})$, an ultrafilter $U_j$ as the collection of sets $X\in L(V_{\lambda+1})\cap\{k:L(V_{\lambda+1})\prec L(V_{\lambda+1})\}$ where $$X\in U_j \Leftrightarrow j\restriction V_\lambda \in jX.$$ Note that $U_j$ is a normal non-principal $L(V_{\lambda+1})$ ultrafilter over $V_{\lambda+1}$, hence the ultrafilter $Ult(L(V_{\lambda+1}), j)=\big(L(V_{\lambda+1}^{\mathcal{E}(V_{\lambda+1})})\cap L(V_{\lambda+1})\big)/U_j$ is well-defined and well-founded. It is important to note that $U_j$ contains only elementary embeddings from $L(V_{\lambda+1})$ to itself which are contructible from $V_{\lambda+1}$ and parameters from this set. As $$I0$$ is therefore equivalent to the existence of a normal non-principle $L(V_{\lambda+1})$ ultrafilter over $V_{\lambda+1}$, the assertion $\kappa$ is $I0$ is $\Sigma_2$ and every critical point of $k: V_{\lambda+2}\prec V_{\lambda+2}$ is $I0$. Unfortunately, this requires $DC_{\lambda}$ to get ultrapower.

An equivalent second-order formulation is: there is some $j:V_\lambda\prec V_\lambda$ and a proper class of ordinals $C$ such that $\alpha_0<\alpha_1<\dots< \alpha_n$ all elements of $C$ and $A\in V_{\lambda+1}$ with $$L_{\alpha_n}(V_{\lambda+1}, \in, \alpha_0, \alpha_1, \dots, \alpha_{n-1})\models \Phi(A)\leftrightarrow \Phi(jA).$$

Similarities with $L(\mathbb{R})$ under Determinacy

The axiom $I0$ was originally formulated by Woodin to establish the consistency of the Axiom of Determinancy. What Woodin established was that $AD^{L(\mathbb R)}$ follows from the existence of an $I0$ cardinal [1]. It is now known that this is a massive overkill; namely, $AD$, $AD^{L(\mathbb R)}$, and infinitely many Woodin cardinals are equiconsistent, and furthermore, $AD^{L(\mathbb R)}$ follows from infinitely many Woodin cardinals with a measurable above them all [1]. This seems like it should be the end of it; $I0$ was simply an axiom to strong for the purpose for which it was created. But there are deeper connections between $AD^{L(\mathbb R)}$ and $I0$.

First off, under $V=L(\mathbb R)$, if $AD$ holds then so does $DC\leftrightarrow DC_\omega$. Similarly, under $I0$ $DC_\lambda$ holds in $L(\mathbb R)$. Furthermore, if $AD$ holds then $\omega_1$ is measurable in $L(\mathbb R)$. Similarly, if $X\subseteq V_{\lambda+1}$ and there is some $j: L(X,V_{\lambda+1})\prec L(X,V_{\lambda+1})$, then $\lambda^+$ is measurable. The connections between $I0$ and determinancy are still not fully understand.[3]

[WIP]

Strengthenings of $\text{I0}$ and Woodin's $E_\alpha(V_{\lambda+1})$ Sequence

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 $\mathrm{crit}(j) < λ$. In particular, “$(V_{λ+1})^{(n+1)♯}$ is Icarus” strongly implies “$(V_{λ+1})^{n♯}$ is Icarus”, but above the first $ω$ sharps it becomes more difficult.[3, 4]

to complete

References

1. Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www   bibtex
2. Kentaro, Sato. Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic 146(2-3):199-236, May, 2007. www   DOI   bibtex
3. Dimonte, Vincenzo. I0 and rank-into-rank axioms. , 2017. arχiv   bibtex
4. Woodin, W Hugh. Suitable extender models II: beyond $\omega$-huge. Journal of Mathematical Logic 11(02):115-436, 2011. www   DOI   bibtex
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