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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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    $\begingroup$ 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 $\beta$. $\endgroup$ Dec 11, 2020 at 7:33
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    $\begingroup$ 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. $\endgroup$ Dec 11, 2020 at 7:45
  • $\begingroup$ Sorry, I misremembered the reflection principle. I have fixed it now. $\endgroup$
    – Master
    Dec 11, 2020 at 16:48
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    $\begingroup$ 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. $\endgroup$ Dec 11, 2020 at 20:40
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    $\begingroup$ I’m skeptical of your composition of embeddings construction. The domains overlap but the functions may not agree on the common domains. $\endgroup$ Dec 11, 2020 at 20:41

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