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I'm looking for a reference (if there is one) for a representation theorem for iterated generic embeddings. What I mean by representation is a generalization of the following:

If $U$ is an ultrafilter on $\kappa$, and we iterate the ultrapower construction to obtain a sequence of embeddings $\langle Ult^{(\alpha)}(V,U),i_{\alpha,\beta} \mid \alpha<\beta<\lambda\rangle$ (for some $\lambda$), then we can define ultrafilters $U_\alpha$ on a subset of $\kappa^\alpha$ (those sequences with finite support) such that $Ult^{(\alpha)}(V,U)\cong Ult(V,U_\alpha)$.

(see e.g. Jech's Third Millennium edition chapter 19).

So I'd like to have a similar representation for iterating generic embeddings (not necessarily using the same forcing at each successor step). This has two ingredients:

  • Generalize the above to iterations where different objects (ultrafilters/extenders) are used to define the embedding in each successor step. Specifically - where the object is not in the image of the previous embedding.
  • Incorporate the forcing mechanism.

Edit: This is the setting I want:

$\langle M_\alpha, i_{\alpha,\beta} \rangle$ is a sequence of models with elementary embeddings $i_{\alpha,\beta}:M_\alpha \to M_\beta$, each $M_\alpha$ defined in a generic extension $N_\alpha$ such that:

  • $N_0=M_0$
  • In successor step: pick a precipitous ideal/tower in $M_\alpha$ and an $N_\alpha$-generc for it. Then $N_{\alpha+1}$ is tye generic extension of $N_\alpha$, and $i_{\alpha,\alpha+1}:M_\alpha \to M_{\alpha+1}$ is the induced embedding.
  • In limit step take the direct limit of the embeddings, and (I hope this works) the finite support iteration of the extensions.

On the face of it it seems feasable, just much more elaborated. Has something like this been done?

[I skimmed the Handbook chapters of Cummings and Foreman, but haven't seen something like this]

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  • $\begingroup$ Isn't Generic ultrapowers by towers (4.8 in Forman's chapter) what you are looking for? $\endgroup$ – Otto Jul 4 at 12:10
  • $\begingroup$ @Otto On the face of it no - there, the whole tower is in the original model, and all the ideals are in that model. My issue is that each step in the iteration is formed using an object which is not in the image of the previous step. Also - I don't necessarily have the coherence required by the definition of a tower. I say "on the face of it" because there might be a way to put what I want into the setting of a tower of ideals, but I don't think it's straightforward. $\endgroup$ – Ur Ya'ar Jul 4 at 12:43
  • $\begingroup$ (by the way the embeddings I'm interested in are in fact induced by towers of ideals, specifically subtowers of the stationary tower) $\endgroup$ – Ur Ya'ar Jul 4 at 13:12
  • $\begingroup$ Thinking a bit more I see that what I wrote isn't completely correct - I do use objects which are in the image of the embedding, but I compute them in the new model, so to form the next step I use functions which were not in the original model. And still I'm not sure I can put everything into a tower, although it might be the case in some examples (perhams using strong enough large cardinal assumptions?). $\endgroup$ – Ur Ya'ar Jul 4 at 13:39
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    $\begingroup$ So you want to alternate between forcing with precipitous ideals and taking the generic embeddings? $\endgroup$ – Asaf Karagila Jul 4 at 20:17
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The following paper does things that may interests you:

Generic large cardinals and systems of filters Giorgio Audrito & Silvia Steila Journal of Symbolic Logic 82 (3):860-892 (2017)

You may also have a look at part V chapter 11 of the draft of my book available here:

http://www.logicatorino.altervista.org/matteo_viale/book.pdf

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  • $\begingroup$ This is very interesting indeed, thanks! Although still I'm not sure my setting fits in, because of the "Compatibility" requirement in these constructions, which I don't seem to have. $\endgroup$ – Ur Ya'ar Jul 9 at 7:03

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