Erdős cardinals and ineffable cardinals

Question

In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have seen proofs in other sources using that Erdős cardinals are subtle, but since Jech does not define subtle cardinals, I wonder whether there is a more direct proof.

Jech proves (Theorem 17.33) that there is an ineffable cardinal below $\eta_\omega$ by constructing an elementary submodel of $V_{\eta_\omega}$ with a set of indiscernibles and a non-trivial elementary embedding. Then the critical point in ineffable. By adding a set of constants to the formal language I can ensure that the critical point is large, so I obtain an unbounded set of ineffable cardinals below $\eta_\omega$ (and the proof works for any $\eta_\alpha$), but how can I ensure a stationary set of ineffable cardinals?

Answer 1

To see that the set of ineffables is stationary, let $C\subseteq \eta_\omega$ be any closed and unbounded set. It suffices to show that the good indiscernibles arising in the definition of $\omega$-Erdos for the structure $(V_{\eta_\omega}, \in, C)$ are in $C$ since by what you have already shown such indiscernibles can be critical points of a non-trivial elementary embedding, and hence are ineffable. However if $\gamma$ is such an indiscernible then $$(V_{\gamma}, \in, C\restriction \gamma)\prec (V_{\eta_\omega}, \in, C).$$

By elementarity, as $C$ is unbounded in $\eta_\omega$, then $C\restriction \gamma$ is unbounded in $\gamma$. That is, $\gamma \in C$.

[ Suppose $\mathcal{A} = \langle L_\delta[A],\in,A,B\rangle $ is a structure. $I\subseteq \delta$ is a good sequence of indiscernibles for $\mathcal{A}$ if for all $\gamma\in I$:

(i) $\langle L_\gamma[A\cap \gamma],\in,A\cap \gamma,B\cap \gamma\rangle \prec \mathcal{A}$;

(ii) $I\backslash \gamma$ is a set of indiscernibles for the structure $\langle L_\delta[A],\in,A,B, \langle \xi\rangle_{\xi < \gamma}\rangle$.

We say that $\delta$ is $\tau$-Erdos if every first order structure $\mathcal{A}=\langle L_\delta[A],\in,A,B\rangle $ has a good sequence of indiscernibles of o.t. $\tau$. It can be shown by a combinatorial argument that each of the $\eta_\alpha$ as defined by Jech are $\alpha$-Erdos. (For this see: J. Baumgartner, Ineffability properties of cardinals II, in: R.E. Butts and J. Hintikka, eds., Logic, Foundations of Mathematics and Computability Theory, Reidel, Dordrecht, 1977, pp 87-106. However it is not the case that $\delta \rightarrow (\tau)^{<\omega}$ implies that $\delta$ is $\tau$- Erdos in general, but the least such $\delta$ (which equals $\eta_\tau$ in Jech's notation) is so.)

An equivalent definition of being $\tau$-Erdos is the following: let $C\subseteq \delta$ be c.u.b. Let $F:[C]^{<\omega}\rightarrow \delta$ be any regressive function (i.e. $F(a)<min(a)$ for all $a\in [C]^{<\omega}$. Then there is a set $I\subseteq\delta$ of o.t.($\tau$) which is homogeneous for $F$: $\forall n \, |F$ " $[I]^n| = 1$.

The notion of good indiscernibility was introduced by Jensen in:

H.-D. Donder, R.B. Jensen, B. Koppelberg, Some applications of K , in: R. Jensen, A. Prestel (Eds.), Set Theory and Model Theory, in: Springer Lecture Notes in Mathematics, vol. 872, Springer-Verlag, 1981, pp. 55–97. ]

The equivalence of the model theoretic notion of $\tau$-Erdos and that with the regressive function partition relation, is somewhat folklorish (it is described by Baumgartner as ``well-known''). The one I have seen is in: A.J. Dodd, The Core Model, London Math. Sot. Lecture Note Series 61 (Cambridge Univ. Press, Cambridge, 1982).