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Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My question is : Is there any reason to collapse only regular cardinals ? What if we collapse singular cardinals, for example the least $\aleph$ fixed point, which has cofinality $\aleph_0$ ? Would it be without interest and why ?

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    $\begingroup$ This seems too vague as stated. Collapses of singular cardinals are used sometimes. It all depends on what kind of properties you are after. Here is an example. $\endgroup$ – Andrés E. Caicedo Oct 29 '19 at 16:37
  • $\begingroup$ (More significant examples appear frequently in the study of determinacy.) $\endgroup$ – Andrés E. Caicedo Oct 29 '19 at 18:29
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    $\begingroup$ The question is extremely confusing because “collapsing” has a standard meaning in the context of forcing, which is how Asaf Karagila understood it, but in fact Jacques is speaking here about ordinal collapsing functions used to define ordinal notations, typically in the context of proof theory. (I know this because the same question was asked by the same username on an entry in my blog which was all about ordinal notations.) $\endgroup$ – Gro-Tsen Oct 29 '19 at 19:52
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    $\begingroup$ …And, of course, the implicit assumption of the question is wrong. The collapsing function for an inaccessible cardinal $I$ gives a name to the smallest fixed point $λ$ of the aleph function (which is much more convenient than this phrase) and is defined at some singular cardinals or non-cardinal ordinals. But it generates $λ$ impredicatively using $I$, so we must compute it there first. $\endgroup$ – Gro-Tsen Oct 29 '19 at 20:02
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    $\begingroup$ @NoahSchweber The blog post is here and the comments are the two signed by “Jacques” (link to comments is at the bottom of the blog post). This is all in French, though (but Google Translate generally does a semi-decent job with my blog posts) and it's more of a popular science (and handwavy) explanation than anything research-level. $\endgroup$ – Gro-Tsen Oct 30 '19 at 16:58
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If $\kappa$ is a singular cardinal and we collapse every ordinal below $\kappa$ to be of size $\lambda$, then you might want to say that $\kappa=\lambda^+$. But $\sf ZFC$ proves that $\lambda^+$ has to be regular, and the fact that $\kappa$ is singular is preserved when going to outer models.

So in effect, we also collapse $\kappa$ itself. In particular, this means that for study of models of $\sf ZFC$ this is not a very useful approach. But we can use them in other ways too, since large enough cardinals have the property that we can force and make them singular (without collapsing them), for example with Prikry, Magidor, or Radin forcings (or their many variants, e.g. diagonal extender-based supercompact Prikry forcing). We can then interleave collapsing functions between the points of the new cofinal sequence, and turn what was once a "very inaccessible cardinal" into $\aleph_\omega$.

This is very useful in the study of combinatorial properties of singular cardinals, as well as their successors (and also for obtaining long sequences of regular cardinals with certain combinatorial properties). And while you're right to claim that this is just combining a few collapses of regular cardinals, this is still something that focuses on singular cardinals as a focal point.


Other than this, in the study of models of $\sf ZF$, collapsing singular cardinals makes sense. Kinda. It turns out that choice plays a significant role in the fact that successor cardinals are regular, and indeed it is consistent that $\omega_1$ is singular, as the results of Feferman–Levy show. This can be extended, thoroughly. Every regular cardinal could have a singular successor of any cofinality, just by repeating the Feferman–Levy construction.

Once we want two successive singular cardinals, though, large cardinals start to play a far more significant role. And everything gets more complicated.

As Andrés Caicedo notes in the comments, though, in the study of models of the Axiom of Determinacy, in some way both of the above uses are combined. We force to collapse the cardinals below a limit of large cardinals to be countable, and then pass to an inner model where the axiom of choice fails. But due to the largeness of the cardinals, it turns out that the singular cardinal which is now $\omega_1$ is actually regular there. Weird, I know.

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In any forcing extension or indeed any kind of extension, if you have collapsed some cardinals, then you have necessarily collapsed cardinals up to some regular cardinal. This is because the first uncollapsed cardinal beyond a given cardinal $\gamma$ will always be regular, because it will be the successor cardinal $\gamma^+$ in the extension, which is regular in the extension and therefore also regular in the ground model.

In this sense, every instance of collapsing is collapsing up to a regular cardinal.

(But meanwhile, I am not claiming that every instance of collapsing is forcing equivalent to the L\'evy collapse up to a regular cardinal, since for example, the forcing to collapse $\omega_1$ in a model of GCH is not isomorphic to the forcing to collapse every ordinal up to $\omega_2$, although they both collapse all the ordinals up to (the ground model) $\omega_2$, in the sense of making all those ordinals countable.)

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    $\begingroup$ +1. To clarify for the OP the statement "in this sense, every instance of collapsing is collapsing up to a regular cardinal" is assuming that the ground model satisfies choice (so there is no tension with Asaf's answer. $\endgroup$ – Noah Schweber Oct 30 '19 at 15:42

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