$\exists j:L(V_{\lambda+1})\to L(V_{\lambda+1})$
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 .
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.
Contents
The $L(V_{\lambda+1})$ Hierarchy
Relation to the I1 Axiom
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.
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
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 that $(V_{λ+1})^{n♯}$ is Icarus, but above the first $ω$ sharps it becomes more difficult.[2, 3]
References
- 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
- Dimonte, Vincenzo. I0 and rank-into-rank axioms. , 2017. arχiv bibtex
- 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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