Consider the language of set theory together with the constant smybols $\langle \Omega_\alpha|\alpha\in Ord\rangle$. Let's add to $ZFC$ the axiom that for every $\alpha$,$n$ there is some $j: V_{\Omega_{\alpha+n}}\prec V_{\Omega_{\alpha+n+1}}$ with critical point $\Omega_\alpha$ and $j: (\Omega_{\alpha+i})=\Omega_{\alpha+i+1}$ for every $i\lt n$. This axiom is one of the strongest reflection principles I know of. It is significantly stronger than $I3$ for instance. However, even this reflection principle is weaker than $I2$.

There are (In my opinion, at least) strong arguments from reflection for $\Pi_n^m$-indescribable, supercompact, extendible, $n$-fold extendible (And by extension $n$-huge), and $I3$. However, even the incredibly strong axiom listed above is weaker than $I2$.

So what kind of reflection principles justify $I2$, $I1$, and $I0$ cardinals? In what way is the assertion that there is some $j: L(V_{\lambda+1})\prec L(V_{\lambda+1})$ a reflection principle? What are the arguements used to justify rank-into-rank axioms.

**Note:** I know about the answer that, due to Gödel's incompleteness theorem, we can never know if these axioms are consistent. I am not looking for answers like that, not because that position has no merit, but because it is obvious and well-known among set-theorists.

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