This question concerns two definitions of the reflection principle. One of them known to be a consequence of the other one. I would like to understand if the reverse is true.

Let us state the first definition of reflection.

Definition(Reflection): A stationary set $S$ in $[H_\lambda]^\omega$ reflects at $X\subseteq H_\lambda$ if $S\cap [X]^\omega$ is stationary in $[X]^\omega$.

The following is the definition of reflection principle which widely is used in literature. See Jech's book, for example.

Definition(Reflection principle) We say ${\rm RP(\lambda)}$ holds if every stationary set of $[H_\lambda]^\omega$ reflects at a set of size $\aleph_1$. We also let ${\rm RP}$ denote ${\rm RP}(\lambda)$ for all regular $\lambda\geq\aleph_2$.

Now we define another definition of reflection:

Definition(Reflection$^*$): A stationary set $S$ in $[H_\lambda]^\omega$ reflects if there is a continuous $\in$-chain $\langle M_\alpha:\alpha<\omega_1\rangle$ of countable elementary substructures of $H_\lambda$ such that $\{\alpha<\omega_1:M_\alpha\in S\}$ is stationary in $\omega_1$. Now using this definition we can build a principle:

Definition(Reflection principle$^*$) We say ${\rm RP}^*(\lambda)$ holds if every stationary subset of $[H_\lambda]^\omega$ reflects. We also let $RP^*$ denote ${\rm RP}^*(\lambda)$ for all $\lambda\geq\aleph_2$.

The following theorem is easy.

Theorem: ${\rm RP}^*$ implies ${\rm RP}$.

Both of these definitions are used and known in literature as reflection principle. Note that ${\rm SRP}$ (Strong reflection principle) implies ${\rm RP}^*$. I would like to know if they are not equivalent.

Question: Is there any model of ${\rm RP}+\neg {\rm RP}^* $?

  • 1
    $\begingroup$ I think you must have meant to add some asterisks in your definition of Reflection* and RP*. Also the $M_\alpha$ should be countable, though technically that follows automatically from what you wrote. $\endgroup$
    – Sean Cox
    Mar 19 '18 at 14:45

The answer is "yes". The principle you call $\text{RP}^*$ is called "reflection to internally club sets" in the literature; as far as I know this terminology first appeared in Foreman-Todorcevic's ``A new Lowenheim-Skolem Theorem". See their paper for the definition of internally stationary, internallly club, and internally approachable.

The main result of Krueger's ``Internal approachability and reflection" gives a model where RP holds---in fact in his model, every stationary subset of any $[H_\lambda]^\omega$ reflects to an internally stationary set of size $\omega_1$---yet there is a stationary set of countable models that doesn't reflect to any internally club set of size $\omega_1$. He doesn't state the main result that way (he states that RP on internally approachable sets fails), but if you look at the proof you'll see that his stationary set that fails to reflect to an internally approachable set, in fact fails to reflect to an internally club set (of size $\omega_1$).

Note that if $\vec{M}$ is a continuous $\in$-chain as in your definition of Reflection*, then its union is an internally club set (i.e. if $W$ is its union, then $W \cap [W]^\omega$ contains a club). And it is straightforward to show that if $S$ is a stationary subset of $[H_\lambda]^\omega$, $S$ reflects to an internally club set of size $\omega_1$ if and only if there exists a continuous $\in$-chain $\vec{M}$ such that $M_\alpha \in S$ for stationarily many $\alpha < \omega_1$.


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