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Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while preserving all cardinals?

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  • $\begingroup$ Very nice question. You will need a measurable cardinal, since the tail filter on the singularizing sequence will construct an inner model with a measurable cardinal. But if $\kappa$ is measurable (and the GCH holds), then you can make it the least inaccessible and then try to use the ground model Prikry forcing. Does this preserve cardinals? I need to think a little more about it. $\endgroup$ Jan 28, 2015 at 13:00
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    $\begingroup$ Joel is right, is it consistent relative to a measurable. You can use the paper of Magidor and Vaananen "On Lowenheim-Skolem-Tarski number for extension of first order logic", where, in the model that they construct, the first inaccessible can change its cofinality to $\omega$ without adding bounded sets or collapsing cardinals. In this paper, they use supercompact in order to get some strong reflection properties. If we don't want those reflection properties, it can be done also from one measurable. $\endgroup$
    – Yair Hayut
    Jan 28, 2015 at 13:07
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    $\begingroup$ @YairHayut Please post an answer giving this argument. $\endgroup$ Jan 28, 2015 at 13:23

1 Answer 1

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In the paper "On Lowenheim-Skolem-Tarski numbers for extension of first order logic", by Magidor and Vaananen, in Theorem 21 they state that it is consistent, relative to the existence of a supercompact cardinal, that the Lowenheim-Skolem-Tarski number of $L(I)$ is the first inaccessible cardinal, were $L(I)$ is the extension of the first order logic with a quantifier of "equicardinality".

An important ingredient in the proof is the following observation (which I state in a slightly different way from the original paper):

Let $\kappa$ be a measurable cardinal. Let $\mathbb{NM}$ be the forcing that adds a club $D$ through the singular cardinals below $\kappa$ using bounded approximations. I will assume that the non limit points of the club are inaccessible cardinals. Let $Col$ be the forcing that collapses all cardinal between the any successor of a point in $D$ and the next point of the $D$, with Easton support. So $\mathbb{NM}\ast Col$ will force $\kappa$ to be the first inaccessible.

On the other hand, in $V$ let $\mathbb{P}$ be a Prikry type forcing that adds a cofinal $\omega$-sequence at $\kappa$, $\{\eta_0, \eta_1, \dots\}$ and pick a sequence of conditions in $\mathbb{NM}$, $\{p_0, p_1,\dots\}$ such that $p_i \subseteq [\gamma_i, \gamma_{i+1})$. So $\mathbb{P}$ singularizes $\kappa$ and adds a club $\tilde{D} = \bigcup p_n$ through the singular cardinals below $\kappa$. Let $\tilde{Col}$ be the forcing that collapses any cardinal between point in $\tilde{D}$ as above, again with Easton support. Note that this time, as $\kappa$ is singular when we define $\tilde{Col}$, the support is unbounded in $\kappa$.

Lemma: $\mathbb{NM}\ast Col$ forces $\kappa$ to be the first inaccessible cardinal. $\mathbb{P}\ast \tilde{Col}$ forces $\text{cf } \kappa = \omega$, and doesn't collapse cardinals above $\kappa$.

Theorem: $\mathbb{P}\ast \tilde{Col}$ adds a generic filter for $\mathbb{NM}\ast Col$, and the quotient forcing, $\mathbb{R}$, changes the cofinaly of $\kappa$ to $\omega$ and doesn't add bounded subsets of $\kappa$.

A variant of this theoerm is proved in the paper of Magidor an Vaananen, during the proof of Theorem 21. The main difference is that they assume there that $\kappa$ is $\kappa^+$-strongly compact in order to obtain a cleaner definition on the generic for $\mathbb{NM}$ that is obtained from the Prikry forcing. Magidor showed me that the same can be achieved using only a measurable, and this is the version that I sketched above.

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  • $\begingroup$ Very interesting. So if I understand the definitions correctly it looks like $\mathbb P * \tilde{Col}$ is not $\kappa^+$-c.c. (because of $\tilde{Col}$). I wonder if we can singularize the least inaccessible $\kappa$ with a $\kappa^+$-c.c. forcing as in the ordinary Prikry forcing. $\endgroup$ Jan 28, 2015 at 15:01
  • $\begingroup$ Yes, you're right. Since $\mathbb{P}\ast \tilde{Col}$ is not $\kappa^+$-c.c., while $\mathbb{NM}\ast Col$ is $\kappa^+$-c.c. (even of size $\kappa$), the quotient forcing $\mathbb{R}$ is not $\kappa^+$-c.c. I don't know if it is possible to singularize the first inaccessible using a $\kappa^+$-c.c. forcing. $\endgroup$
    – Yair Hayut
    Jan 28, 2015 at 15:15
  • $\begingroup$ Yair, what about homogeneous forcing? :-) $\endgroup$
    – Asaf Karagila
    Jan 28, 2015 at 15:16
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    $\begingroup$ @MonroeEskew: Maybe the paper of Gitik "Changing cofinalities and the non-stationary ideal" will be useful for this question. Theorem II in this paper states that it's consistent relative to supercompact that for some inaccessible $\kappa$ there is stationary $S\subseteq S^\kappa_\omega$ such that $NS_\kappa \restriction S$ is saturated, so forcing a generic ultrafilter will force $\text{cf }\kappa = \omega$, won't add bounded subsets and will be $\kappa$-c.c. I think that it's possible to arrange that $\kappa$ is the first inaccessible in this model. $\endgroup$
    – Yair Hayut
    Jan 28, 2015 at 15:56

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