Set-theoretic geology III: inside the core

Question

Thanks to Jonas, Asaf, and Gabe I understand a little more of grounds and the mantle (or mantles, because it looks like there may be more than one).

But, set-theoretic geology, or so it seems to me, should not be exclusively about grounds: the Earth has a core, and in fact perhaps several strata of cores, till it reaches the mantle (or some intermediate zone).

So, how about doing the opposite? Rather than penetrating a model $M$ by erosion, what if we know that there is a solid center ( say $L^M$, assuming that the model is not $L[G]$ for some generic set), and grow that minimal core as far as we can go?

In other words, let us define a class of M-non grounds,

$$\mathrm{CORE}= \{ N\ \nsubseteqq M, N \vDash ZFC \land \nexists G M=N[G] \}$$

and determine its structure. In some cases this class is empty, but suppose it is not.

Question:

what can be said of the non grounds of $M$? Is this class a upward directed partial order? When do this class reach the mantle (in the sense that I cannot find anything below the mantle that is not in CORE)?

Are there any layers for some models which lie between the outer core and the intersection of all mantles?

Sounds like Jules Verne's Voyage au Centre de la Terre $\dots$

Answer 1

Here are a few observations about CORE.

Claim: It is consistent that CORE is not pairwise upwards directed.

Proof: Let $\mathbb{P}_0$ be the class forcing for the Easton product of the Cohen forcing $\mathrm{Add}(\alpha^+, 1)$, over all cardinals $\alpha$, in $L$. Let $G$ be $L$-generic for $\mathbb{P}_0$. Then, we can pick a partition $A \cup B$ of the class of cardinals, with both $A, B$ being proper classes. Then $N_0 = L[G \restriction A], N_1 = L[G \restriction B]$ are both in CORE, and have no common upper bound in CORE.

On the other hand:

Claim: It is consistent that CORE is upwards directed:

Proof: Let $\mathbb{P}_1$ be the backwards Easton support iteration of $\mathrm{Add}(\alpha^+, 1)$ for $\alpha$ successor of a singular cardinal in $L$. Let $G$ be an $L$-generic and let $M=L[G]$.

Sub Claim: For every $N \in CORE^M$, there is an ordinal $\alpha$ such that $N \subseteq L[G \restriction \alpha]$. In particular, CORE is upwards directed.

Sketch of Proof: Let $\alpha$ be minimal (successor of singular) such that $L[G\restriction \alpha] \not\subseteq N$ and let $x\in N$ be a set of ordinals of minimal rank such that $x \notin L[G \restriction \alpha]$. Then $x$ has to be a fresh set over $L[G \restriction \alpha]$, and by minimality if has to be a subset of $\alpha^+$. By gap-type arguments, $x$ together with its name (which is in $L$) codes the set $G \restriction \alpha$, and thus $L[G \restriction \alpha] \subseteq N$.

Finally, large cardinals seems to have negative affect on the directedness of CORE:

Claim: Let $\kappa \in M$ measurable and $2^\kappa = \kappa^{+}$. Then, there are $N_0, N_1 \in CORE^M$, and $x \in N_1$, $y \in N_0$, such that $N_0[x] = N_1[y] = M$.

Proof: Let $\mathcal{U}$ be a normal ultrafilter on $\kappa$ and let $N$ the ultrapower by $\mathcal{U}$. Let us construct inside $M$ two $N$-generic filters $G_0, G_1 \subseteq \mathrm{Add}(\kappa^{+}, 1)$, such that their bitwise xor codes $\mathcal{U}$. This is possible, since $2^\kappa = \kappa^{+}$ (both in $M$ and in $N$). Let $N_0 = N[G_0], N_1 = N[G_1]$ and note that $N_0[G_1] = N_1[G_0] \supseteq N[\mathcal{U}] = M$. QED

On the other hand, if $A$ is a set of ordinals and $A^{\#}$ exists, then $L[A] \in CORE$, since no set forcing in $L[A]$ can introduce a sharp for $A$. So under the large cardinal axiom "every set has a sharp" (which follows from the existence of class of measurable cardinals, for example), $\bigcup CORE = V$, so CORE can certainly contain sets which are not in the mantle.