The reflection principle $R_2$

A few years ago, in this paper, I introduced the following reflection principle:

Definition. $R_2(\theta ,\kappa )$ asserts that for every function $f:E_{<\kappa }^{\theta }\to \kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta \in E_{\kappa }^{\theta }\mid f^{-1}[j]\cap \delta \text{ is nonstationary}\}.$$

I wrote there that by a theorem of Magidor, $R_2({\mathrm{\aleph }}_2,{\mathrm{\aleph }}_1)$ is consistent modulo the existence of a weakly compact cardinal, and at the end of that paper, I asked (Question 3) what is the consistency strength of $R_2({\mathrm{\aleph }}_2,{\mathrm{\aleph }}_1)$.
People I asked about this mentioned Magidor’s other result that if there exist two stationary subset of $E_{{\mathrm{\aleph }}_0}^{{\mathrm{\aleph }}_2}$ that do not reflect simultaneously, then ${\mathrm{\aleph }}_2$ is weakly compact in $L$, however, to address $R_2({\mathrm{\aleph }}_2,{\mathrm{\aleph }}_1)$, a more complicated counterexample is needed.

In this post, I will answer my own question, proving that the consistency strength of $R_2({\mathrm{\aleph }}_2,{\mathrm{\aleph }}_1)$ is exactly that of a weakly compact cardinal.

Proposition 1. If $R_2({\mathrm{\aleph }}_2,{\mathrm{\aleph }}_1)$ holds, then ${\mathrm{\aleph }}_2$ is weakly compact in $L$.
Proof. As mentioned in a previous blog post, if ${\mathrm{\aleph }}_2$ is not weakly compact in $L$, then $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}({\mathrm{\aleph }}_2)$ holds. Now, appeal to the next proposition. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◼"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fc.svg">}$

Proposition 2. If $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}(\theta )$ holds, then $R_2(\theta ,\kappa )$ fails for every regular uncountable cardinal $\kappa <\theta$.
Proof. Let $\kappa <\theta$ be arbitrary regular uncountable cardinals. By Lemma 3.2 of this paper, we may fix a sequence $⟨C_{\delta }\mid \delta <\theta ⟩$ such that:

• $C_{\delta }$ is a club in $\delta$ for all limit $\delta <\theta$;
• if $\alpha \in \text{acc}(C_{\delta })$, then $C_{\alpha }=C_{\delta }\cap \alpha$;
• $S_{\gamma }:=\{\delta \in E_{\kappa }^{\theta }\mid min(C_{\delta })=\gamma \}$ is stationary for all $\gamma <\theta$.

(here, $\text{acc}(C)=\{\alpha \in C\mid sup(C\cap \alpha )=\alpha >0\}$.)

Now, define $f:E_{<\kappa }^{\theta }\to \kappa$ by stipulating:$$f(\alpha ):=\{\begin{array}{ll}min(C_{\alpha }),& \text{if }min(C_{\alpha })<\kappa \\ 0,& \text{otherwise.}\end{array}$$

Finally, let $j<\kappa$ be arbitrary. To prove that $A_j$ is not nonstationary, let us show that it contains the stationary set $S_j$.

Towards a contradiction, suppose that $\delta \in S_j\setminus A_j$. Then $f^{-1}[j]\cap \delta$ is stationary, and we may pick some $\alpha \in f^{-1}[j]\cap \text{acc}(C_{\delta })$.

• By $\alpha \in f^{-1}[j]$, we have either $min(C_{\alpha })=f(\alpha )<j$ or $min(C_{\alpha })\ge \kappa$.
• By $\alpha \in \text{acc}(C_{\delta })$ and $\delta \in S_j$, we have $min(C_{\alpha })=min(C_{\delta }\cap \alpha )=min(C_{\delta })=j$.

This is a contradiction. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◼"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fc.svg">}$