Gitik’s Long Extender Forcing – Preliminaries

I decided to work back through the presentation of Gitik’s long extender forcing from his Handbook chapter, and I’m using this blog to motivate me to actually type stuff out. Unfortunately, I will probably stop updating this blog at some point again because I don’t exactly like WordPress as a medium for a math blog. However, as long as I feel like updating this thing, I feel like I should make use of it. This post will assume knowledge of the basic Prikry forcing as well as the diagonal Prikry forcing (mainly for motivational purposes), presentations of which can be found in Gitik’s handbook chapter as well as here. Actually, this post was part of the reason I ended up posting my unfinished notes from Spencer’s talk.

The diagonal Prikry forcing uses a sequence of measurable cardinals \langle \kappa_n : n<\omega\rangle and witnessing measures to add a Prikry sequence to \prod_{n<\omega}\kappa_n dominating all ground model functions. So, in order to blow up the power of \kappa=\sup_{n<\omega}\kappa_n, we just have to find a way of doing this diagonal Prikry forcing a bunch of times in a nice and coherent way. Since we used measures on each \kappa_n to accomplish this, the natural step to take is to utilize extenders of the appropriate length, since those naturally carry a bunch of measures that are associated in a nice way. The hope is that we can generalize the diagonal Prikry forcing construction using extenders, and have nothing horrible go wrong (like accidentally collapsing cardinals). It turns out that this approach almost works. I want to use this post to flesh out this “naive” idea, and then to present Gitik’s actual construction.

The presentation in this post is based off of the presentation in Gitik’s Handbook Chapter, as well as Gitik and Unger’s notes from the Appalachian Set Theory Workshop. We will assume that GCH holds in the ground model.

Suppose that we have an increasing sequence of \lambda-strong cardinals \langle \kappa_n : n<\omega\rangle where \kappa=\sup_{n<\omega} \kappa_n, and \lambda\geq\kappa^{++}. For each \kappa_n, fix a non-trivial, elementary embedding j_n:V\to M with the following properties:

  1. crit(j_n)=\kappa_n;
  2. j_n(\kappa_n) > \lambda;
  3. V_{\lambda+1}\subseteq M, and {}^{\kappa_n}M\subseteq M.

For n<\omega, and \kappa_n\leq \alpha <\lambda, define E_{n\alpha}=\{X\subseteq \kappa_n : \alpha \in j(X)\}. Note that each E_{n\alpha} is a \kappa_n-complete, non-principal ultrafilter on \kappa_n with E_{n\kappa_n} normal. These are the measures that we will be working with. The presence of these \lambda-strong embeddings will give us a nice \kappa_n-directed order on these measures, which will allow us to perform the diagonal Prikry forcing \lambda-many times.

For \alpha\leq \beta<\lambda, we say that \alpha\leq_{E_n}\beta if there is some f:\kappa_n\to\kappa_n such that j_n(f)(\beta)=\alpha. This extender ordering allows us to project the ultrafilter E_{n\beta} to E_{n\alpha}. To see this, assume we have that \alpha\leq_{E_n}\beta, and let f be such that j_n(f)(\beta)=\alpha. Let A\in E_{n\beta}, and note that by definition we have \beta\in j(A). By elementarity, we then have that \alpha=j_n(f)(\beta)\in j_n(f''j_n(A)), but again by elementarity we know that j_n(f''j_n(A))=j_n(f'' A). Thus, we have that \alpha\in j_n(f''A) and so by definition f''A\in E_{n\alpha}. We now fix some notation.

  • If \alpha\leq_{E_n}\beta, we will suggestively call the witnessing function \pi_{\beta\alpha}. That is, we have j(\pi_{\beta\alpha})(\beta)=\alpha.
  • Let N_\alpha=\mathrm{Ult}(V,E_{n\alpha}), and let i_\alpha : V\to N_\alpha be the associated ultrapower embedding.
  • The general theory of ultrapowers gives us a factor map k_\alpha : N_\alpha \to M_n, given by k([f]_{E_{n\alpha}})=j(f)(\alpha).

Note that if \beta\leq_{En}\alpha, we have that \beta=j(\pi_{\alpha\beta})(\alpha)=k_\alpha([\pi_{\alpha\beta}]_{E_{n\alpha}}). Let \beta^*=[\pi_{\alpha\beta}]_{E_{n\alpha}}, and note that this is, roughly speaking, what N_\alpha things the seed of E_{n\beta} is. We can then use this ordinal to define an elementary embedding k_{\beta\alpha}: N_\beta\to N_\alpha that makes some awful looking diagram commute. Basically, this ordering yields a directed system of ultrapowers and their associated embeddings, the direct limit of which is the embedding j_n: V \to M_n. Of course, here we’re brushing a lot of technical details under the rug, but this is the basic idea.

We are not so concerned with the directed system of embeddings and ultrapowers itself. We are much more concerned with how the E_n ordering relates the measures themselves. The following three lemmas are somewhat technical, but relatively standard affair with arguments involving commutative diagrams of elementary embeddings.

Lemma: (GCH) Fix n<\omega, and \tau<\kappa_n. If \{ \alpha_\nu : \nu<\tau\} is a collection of ordinals below \lambda, then there are \lambda-many \alpha<\lambda such that \alpha\geq_{E_n}\alpha_\nu for each \nu<\tau. In particular, \leq_{E_n} is \kappa_n-directed.

Proof: Since GCH holds, we can enumerate [\kappa_n]^{<\kappa_n} in order type \kappa_n by \langle a_\beta : \beta<\kappa_n\rangle with the property that, for any regular \delta<\kappa_n, \langle a_\beta : \beta<\delta\rangle enumerates [\delta]^{<\delta} such that every x\in[\delta]^{<\delta} appears \delta-many times. Now we want to look at j(\langle a_\beta :\beta<\kappa_n\rangle), and note that since j(\kappa_n)>\lambda, by elementarity we have that j(\langle a_\beta :\beta<\kappa_n\rangle)\upharpoonright \lambda enumerates [\lambda]^{<\lambda} in order type \lambda with the above property. Call this enumeration \langle a_\beta :\beta<\lambda\rangle. Fix some \alpha<\lambda such that a_\alpha=\langle \alpha_\nu : \nu<\tau\rangle, we claim that \alpha\geq_{E_n}\alpha_\nu for each \nu<\tau. Since there are \lambda-many such ordinals \alpha, that will complete the proof.

By standard ultrapower stuff, we have that \langle a_\beta : \beta<\lambda\rangle= j(\langle a_\beta :\beta<\kappa_n\rangle)=k_\alpha(i_\alpha(\langle a_\beta :\beta<\kappa_n \rangle)). Since a_\alpha is the \alphath member of the above sequence, we can express a_\alpha as a_\alpha=k_\alpha(i_\alpha(\langle a_\beta :\beta<\kappa_n \rangle)([id]_{E_{n\alpha}})).  Since crit(k_\alpha)\geq crit(j_n)=\kappa_n, it follows that i_\alpha(\langle a_\beta :\beta<\kappa_n \rangle)([id]_{E_{n\alpha}}) is a \tau-length sequence of ordinals from N_\alpha. Fix \nu<\tau, and let \alpha_\nu^* be the \nuth member of this sequence, and note that k_\alpha(\alpha_\nu^*)=\alpha_\nu by elementarity. As before, we’ve isolated what N_\alpha thinks is the seed of E_{n\alpha_\nu}, and we can use this to define an embedding from N_{\alpha_\nu} \to N_\alpha as well as a projection map \pi_{\alpha,\alpha_\nu}.

We define k_{\alpha_\nu,\alpha}:N_{\alpha_\nu}\to N_\alpha by k_{\alpha_\nu,\alpha}([f]_{E_{n\alpha_\nu}})=i_\alpha(f)(\alpha_\nu^*). It is fairly routine to check that k_{\alpha_\nu,\alpha} is elementary and commutes with i_\alpha and i_{\alpha_\nu}. Next, let \pi_{\alpha,\alpha_\nu}:\kappa_n\to\kappa_n be such that [\pi_{\alpha,\alpha_{\nu}}]_{E_{n\alpha_\nu}}=\alpha_\nu^*. By definition, j(\pi_{\alpha,\alpha_\nu})(\alpha)=k_\alpha([\pi_{\alpha,\alpha_\nu}]_{E_{n\alpha}})=k(\alpha_\nu^*)=\alpha_\nu, and thus \alpha_\nu\leq_{E_n} \alpha. Since \nu<\tau was arbitrary, this completes the proof.

Now that we’ve shown that the extender order is \kappa_n-directed, we will show that \leq_{E_n} has some nice coherency properties that will allow us to generically add \lambda-many sequences to \prod_{n<\omega}\kappa_n.

Lemma: Suppose that \gamma<\beta\leq \alpha<\lambda. If \beta\leq_{E_n}\alpha and \gamma\leq_{E_n}\alpha, that \{\nu<\kappa : \pi_{\alpha\beta}(\nu)<\pi_{\alpha\gamma}(\nu)\}\in E_{n\alpha}.

Proof: This is relatively simple, as we simply need to show that N_\alpha\models [\pi_{\alpha\beta}]_{E_{n\alpha}}<[\pi_{\alpha\gamma}]_{E_n\alpha}. But we’ve already shown that k_\alpha([\pi_{\alpha\beta}]_{E_{n\alpha}})=\beta and k_\alpha([\pi_{\alpha\gamma}]_{E_{n\alpha}})=\gamma, and thus it follows by elementarity and the fact that M\models \beta<\gamma that N_\alpha\models [\pi_{\alpha\beta}]_{E_{n\alpha}}<[\pi_{\alpha\gamma}]_{E_n\alpha} and so the lemma is proved.

Lemma: Suppose that we have ordinals \gamma\leq\beta\leq\alpha<\lambda such that \gamma\leq_{E_n}\beta\leq_{E_n}\alpha, then there is some A\in E_{n\alpha} such that, for each \nu\in A, \pi_{\alpha\gamma}(\nu)=\pi_{\beta\gamma}(\pi_{\alpha\beta}(\nu)).

Proof: Note that we only need to show that [\pi_{\alpha\gamma}]_{E_{n\alpha}}=[\pi_{\beta\gamma}\circ\pi_{\alpha\beta}]_{E_{n\alpha}}. We already know that k_\alpha([\pi_{\alpha\gamma}]_{E_{n\alpha}})=\gamma, so by the above argument, we only need to show that k_\alpha([\pi_{\beta\gamma}\circ\pi_{\alpha\beta}]_{E_{n\alpha}})=\gamma. By definition and liberal applications of elementarity, we have:

k_\alpha([\pi_{\beta\gamma}\circ\pi_{\alpha\beta}]_{E_{n\alpha}})=j_n(\pi_{\beta\gamma}\circ\pi_{\alpha\beta})(\alpha)=j_n(\pi_{\beta\gamma})(j_n(\pi_{\alpha\beta}(\alpha))=j_n(\pi_{\beta\gamma})(\beta)=\gamma.

This completes the proof.

With these properties of the extender ordering in hand, we can describe our first approach to adding \lambda-many sequences to \prod_{n<\omega}\kappa_n using the presence of (\kappa_n, \lambda+1) extenders. The idea is, for each \kappa_n, to use elements a\in[\lambda]^{<\kappa_n} such that each a contains a \leq_{E_n}-maximal element \max(a), and do the diagonal Prikry forcing at the sequence of ultrafilters corresponding to these maximal elements.  This will yield a Prikry sequence through \prod_{n<\omega}\kappa_n with each Prikry point at each ultrafilter. The properties extender order will then allow us to project this Prikry point nicely through each element of a, giving us many \omega-sequences. But, we can use the coherency properties of this order to move the sequences a around, and therefore generically cover \lambda in this manner. We now attempt to make this idea more precise.

Definition: We let \mathbb{P}^* be the forcing consisting of conditions of the form (s_0,s_1,\ldots, s_{n-1}, \langle a_n, A_n\rangle, \langle a_{n+1}, A_{n+1}\rangle,\ldots) where:

  1. For m<n, s_m\subseteq \kappa_n.
  2. For each n\leq m<\omega, a_m\in[\lambda]^{<\kappa_m}.
  3. Each a_m has a \leq_{E_m}-maximal element \max(a).
  4. For each m\geq n, A_m\in E_{m,\max(a)}.
  5. For each \alpha,\beta,\gamma\in a_m, if \alpha\geq_{E_m}\beta\geq_{E_m}\gamma, then \pi_{\alpha\gamma}(\nu)=\pi_{\beta\gamma}(\pi_{\alpha\beta}(\nu)) for each \nu\in \pi_{\max(a)\alpha}{} ''A_m\in E_{m\alpha}.
  6. For each \alpha>\beta \in a_m, and for each \mu\in A_m, \pi_{\max(a)\alpha}(\nu)=\pi_{\max(a)\beta}(\nu).

In some sense, this is the natural definition for this forcing given what we know about the extender ordering. Unfortunately, it’s rather complicated compared to the diagonal Prikry forcing. We now define the ordering by saying that (s_0,\ldots, s_{n-1}, \langle a_n, A_n\rangle, \ldots)\leq (t_0,\ldots, t_{m-1}, \langle a_m, A_m\rangle, \ldots) if the following hold:

  1. m\leq n;
  2. For i<m, s_i=t_i;
  3. For i\in [m,n), $s_i$ is a collection of ordinals in members of ultrafilters corresponding to the sequence b_i;
  4. For n\leq i, a_i\supseteq b_i;
  5. For n\leq i, \pi_{\max(a),\max(b)}{}''A\subseteq B_i.

I’m being a bit imprecise with the third condition, but the idea’s there. It should be noted that, unlike the case for diagonal Prikry forcing, conditions with the same stems in the ordering we’ve constructed may have many incompatible extensions. So, we have no easy way of showing that cardinals above \kappa are preserved, and it turns out that forcing with \mathbb{P}^* will actually collapse cardinals. So, we need to tweak this idea a bit and add more structure in order to keep this collapse from happening. In the next post, I will present Gitik’s definition of his Long Extender forcing, which is actually a small adjustment of this above idea, except he uses Cohen conditions to add some extra structure and hide these Prikry sequences.

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