October 7
Sakaé Fuchino, Kobe University
Definability of Laver-generic large cardinals and largeness of generic large cardinals with chain conditions

For a class ${\mathcal{P}}$ of posets, a cardinal $\kappa$ is said to be generically supercompact by ${\mathcal{P}}$ (or ${\mathcal{P}}$-gen. supercompact for short) if, for any $\lambda\geq\kappa$, there are $P\in{\mathcal{P}}$ such that, for all $({\sf V},P)$-generic $G$ there are $j$, $M\subseteq{\sf V}[G]$ with $j:{{\sf V}}\stackrel{\prec}{\rightarrow}_\kappa{M}$, $j(\kappa)\gt\lambda$, and $j\mbox{''}{\lambda}\in M$.

A cardinal $\kappa$ is Laver-generically supercompact for ${\mathcal{P}}$ (or ${\mathcal{P}}$-Laver-gen. supercompact for short) if, for any $\lambda\geq\kappa$, $P\in{\mathcal{P}}$ and $({\sf V},P)$-generic $G$, there are ${\mathcal{P}}$-name $\dot{Q}$ with $\Vdash_{P}\,''\dot{Q}\in{\mathcal{P}}''$ such that, for all $({\sf V},P*\dot{Q})$-generic $H \supseteq G$, there are $j$, $M\subseteq{\sf V}[H]$ such that $j:{{\sf V}}\stackrel{\prec}{\rightarrow}_\kappa{M}$, $j(\kappa)\gt\lambda$, and $P*\dot{Q}$, $H$, $j\mbox{''}\lambda\in M$.

$\mathcal P$-gen. superhuge, and $\mathcal P$-Laver-gen. superhuge cardinals are defined if the condition $j\mbox{''}\lambda\in M$ is replaced with $j\mbox{''}j(\kappa)\in M$.

Perhaps it is not apparent at first sight in the formulation the definitions above but these notions of generic large cardinals are first-order definable (S.F, and H. Sakai [1]).

While the generic supercompactness does not determine the size of the cardinal. Laver-generic supercompactness determines the size of the cardinal and that of the continuum in most of the natural settings of ${\mathcal{P}}$ (see S.F., A.Ottenbreit Maschio Rodrigues, and H. Sakai [0] for a proof):

(A) If $\kappa$ is ${\mathcal{P}}$-Laver-gen. supercompact for a class ${\mathcal{P}}$ of posets such that (1) all $P\in{\mathcal{P}}$ are $\omega_1$-preserving, (2) all $P\in{\mathcal{P}}$ do not add reals, and (3) there is a $P_1\in{\mathcal{P}}$ which collapses $\omega_2$, then $\kappa=\aleph_2$ and CH holds.

(B) If $\kappa$ is ${\mathcal{P}}$-Laver-gen. supercompact for a class ${\mathcal{P}}$ of posets such that (1) all $P\in{\mathcal{P}}$ are $\omega_1$-preserving, (2)' there is an a $P_0\in{\mathcal{P}}$ which add a real, and (3) there is a $P_1$ which collapses $\omega_2$, then $\kappa=\aleph_2=2^{\aleph_0}$.

(C) If $\kappa$ is ${\mathcal{P}}$-Laver-gen. supercompact for a class ${\mathcal{P}}$ of posets such that (1)' all $P\in{\mathcal{P}}$ preserve cardinals, and (2)' there is a $P_0\in{\mathcal{P}}$ which adds a real, then $\kappa$ is very large and $\kappa\leq 2^{\aleph_0}$.

The case (C) can be still improved ([0]):

(C') If $\kappa$ is tightly ${\mathcal{P}}$-Laver-gen. superhuge for a class ${\mathcal{P}}$ of posets such that (1)' all $P\in{\mathcal{P}}$ preserve cardinals, and (2)' there is a $P_0\in{\mathcal{P}}$ which adds a real, then $\kappa$ is very large and $\kappa=2^{\aleph_0}$.

(A ${\mathcal{P}}$-Laver-gen. superhuge cardinal $\kappa$ is tightly ${\mathcal{P}}$-Laver-gen. superhuge, if $\dot{Q}$ in the definition of Laver-gen. superhugeness can always be chosen to be small enough --- see [0] for a precise definition.)

In this talk, we are going to give a sketch of the proof of definability and discuss about a theorem which assesses the largeness of $\kappa$ in (C) under the additional assumption that elements of ${\mathcal{P}}$ satisfy certain chain conditions.

[0] S.F., A.Ottenbreit Maschio Rodrigues, and H.Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, II --- reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495--523. https://fuchino.ddo.jp/papers/SDLS-x.pdf
[1] S.F., and H.Sakai, Generically supercompact cardinals by forcing with chain conditions RIMS Kôkûroku, No.2213 (2022). https://fuchino.ddo.jp/papers/RIMS2021-ccc-gen-supercompact-x.pdf
[2] S.F., and H.Sakai, The first-order definability of generic large cardinals, to appear. https://fuchino.ddo.jp/papers/definability-of-glc-x.pdf

Video