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- The $L(V_{\lambda+1})$ Hierarchy
- Relation to the I1 Axiom
- Ultrapower Reformulation
- Similarities with $L(\mathbb{R})$ under Determinacy
- Strengthenings of $\text{I0}$ and Woodin’s $E_\alpha(V_{\lambda+1})$ Sequence

*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 (Kanamori, 2009).

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” (Kanamori, 2009).
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.

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 \(L\_{\\alpha\_n}(V\_{\\lambda+1}, \\in, \\alpha\_0, \\alpha\_1, \\dots, \\alpha\_{n-1})\\models \\Phi(A)\\leftrightarrow \\Phi(jA).\)

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 (Kanamori, 2009). 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 (Kanamori, 2009). 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.(Dimonte, 2017)

[WIP]

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. (Dimonte, 2017; Woodin, 2011)

*to complete*

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- Kanamori, A. (2009).
*The higher infinite*(Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3 - Dimonte, V. (2017).
*I0 and rank-into-rank axioms*. - Woodin, W. H. (2011). Suitable extender models II: beyond ω-huge.
*Journal of Mathematical Logic*,*11*(02), 115–436. https://doi.org/10.1142/S021906131100102X