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?

  • 1
    $\begingroup$ @Tomek Kania Thank you for correcting the double accent in "Erdős". I have just corrected a third occurence you missed. I apologize for my unfamiliarity with the Hungarian spelling. $\endgroup$
    – Carlos
    Aug 30 '14 at 12:45

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).

  • $\begingroup$ Philip, could you add the definition of good indiscernibles? I think not everyone is familiar with them. $\endgroup$ Aug 29 '14 at 17:54
  • $\begingroup$ I wrote the CA entry and wondered for a few minutes this morning whether I made a mistake :). But then I remembered about good indiscernibles and was going to write it up later. $\endgroup$ Aug 29 '14 at 17:56
  • $\begingroup$ @Victoria: I remember that one morning I was going to remind myself to do something later; I still can't remember what was that thing and whether I did it or not. :-) $\endgroup$
    – Asaf Karagila
    Aug 29 '14 at 18:03
  • $\begingroup$ @Philip Welch Thank you for your answer! I did not know the concept of good indiscernible but I have found in the web one of your articles (Greatly Erdös Cardinals...) with the definition. So I understand your answer, but my problem now is that in your paper you define Erdös cardinals as those for which there are good indiscernibles in each structure. So I would need a proof that Erdös cardinals so defined are the same than those defined by the usual partition relation. Could you give me a reference? Thanks again. $\endgroup$
    – Carlos
    Aug 29 '14 at 20:32
  • $\begingroup$ @Carlos Edited to add refs. for your query. $\endgroup$ Aug 30 '14 at 10:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.