# Reflection principles justifying $I2$ and larger cardinals

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.

• Why is this stronger than I3? I’m pretty sure it’s weaker than a 2-huge. Apply Rowbottom’s theorem to the measure derived from a 2-hugeness embedding to get a measure one set $X$ such that for all $\alpha<\beta$ in $X$, there is a hugeness ultrafilter with critical point $\alpha$ and target $\beta$. Dec 11, 2020 at 7:33
• Correction: It’s weaker than $\kappa$ that is $2^{\kappa}$-supercompact. Let $j : V \to M$ witness this supercompacntess with critical point $\kappa$. $M$ can see $j \restriction V_{\kappa+1}$. Let $U$ be the measure on $\kappa$ derived from $j$. Reflect to get $A \in U$ such that for all $\alpha \in A$, there is $j_\alpha : V_{\alpha+1} \to V_{\kappa+1}$, with critical point $\alpha$, $j(\alpha)=\kappa$. Reflect again and use diagonal intersection to a measure one set $B \subseteq A$ witnessing your reflection principle. Dec 11, 2020 at 7:45
• Sorry, I misremembered the reflection principle. I have fixed it now. Dec 11, 2020 at 16:48
• Use two colors on the finite subsets like I described. Rowbottom proved that measurable cardinals are Ramsey, which means there is a measure one set such that got all $n$, every size $n$ set gets the same color. Dec 11, 2020 at 20:40
• I’m skeptical of your composition of embeddings construction. The domains overlap but the functions may not agree on the common domains. Dec 11, 2020 at 20:41