text:
\section{Introduction}

\subsection{Steel's Programme}

In \cite{steel2014}, John Steel presented a version of the set-theoretic multiverse consisting of `worlds' (\textit{set-generic extensions} of $V$), and a collection of axioms for it, $\mathsf{MV}$, and also made the hypothesis that such a multiverse might contain a \textit{core}, that is, a world included in all other worlds, which would act as `preferred universe', as the `real $V$'.\footnote{\cite{steel2014}, p. 167ff. Further discussion of Steel's $\mathsf{MV}$ axioms may be found in \cite{maddy-meadows2020} (for the discussion of the `core hypothesis', see, in particular, sections 5-6).}  A few years after the appearance of \cite{steel2014}, in the context of research on \textit{set-theoretic geology},\footnote{For a review of some set-theoretic geological concepts, see section 3.1.} it was proved (in \cite{U:ECM}) that, if there exists an \textit{extendible} cardinal, then $V$ has a smallest ground, and that such smallest ground is the $\kappa$-mantle of $V$ itself, where $\kappa$ is the least   extendible cardinal. A noticeable consequence of Usuba's result is that the theory $\mathsf{MV}_T$, where $T=\mathsf{ZFC}+$`there exists a proper class of extendible cardinals' has a \textit{core}. Now, given the presuppositions of $\mathsf{MV}$, in particular, its reliance upon the whole hierarchy of Large Cardinals (LCs), it makes full sense to investigate the features of the theory $\mathsf{MV}_T$, in particular, the features of its core.

Steel's `core hypothesis' also has connections with another, recently arisen, fundamental set-theoretic hypothesis, that is, Woodin's `Ultimate-$L$ Conjecture'.\footnote{\label{Woodin}For the bulk of all results on Ultimate-$L$, see \cite{woodin2017}. For a more accessible exposition, see \cite{woodin2011b}, pp. 457-69. See also section 5 of the present paper.} The inner model programme has progressively unveiled the existence of `canonical' inner models of $\mathsf{ZFC}$+LCs; Woodin's Ultimate-$L$ Conjecture asserts that $\mathsf{ZFC}$+LCs also proves the existence of a \textit{weak extender model} for a cardinal $\delta$ being supercompact which also satisfies $V$=Ultimate-$L$.\footnote{\label{supercomp} An uncountable cardinal $\kappa$ is $\gamma$-supercompact, for some $\gamma \in Ord$,  if and only if there is an elementary embedding $j: V \rightarrow M$, with critical point $\kappa$, such that $M^{\gamma} \subseteq M$. A supercompact cardinal $\kappa$ is a cardinal that is $\gamma$-supercompact for all $\gamma \in Ord$.} In turn, Steel has made the hypothesis that Ultimate-$L$ might be the most `suitable' candidate as the \textit{core} of $\mathsf{MV}$.\footnote{\cite{steel2014}, p. 169ff., in particular, section 6. See also footnote \ref{AxiomH}, and \cite{maddy-meadows2020}, pp. 46-47.}

Now, given $\mathsf{MV}_T$, where, again, $T=\mathsf{ZFC}$+`there exists a proper class of extendible cardinals', Steel's Programme, as we will call it, may be formulated as follows:

\vspace{11pt}

\noindent
\textbf{Steel's Programme}. Use facts about the core of $\mathsf{MV}_T$ as evidence for the following claims:

\begin{enumerate}

\item $V$ is the core.

\item The core is Ultimate-$L$.

\item $\mathsf{ZFC}$+LCs+$V$=Ultimate-$L$ has a better claim than any other theory to be seen as the best (ultimate) extension of $\mathsf{ZFC}$.\footnote{\label{steel}It should be noted that Steel's Programme, although `implicit' in \cite{steel2014}, is not openly formulated as such by Steel himself.}

\end{enumerate}

\noindent
As we shall see in more detail later on, the execution (and meaningfulness) of Steel's Programme crucially depends on the relationship between $\mathcal{L}_\mathsf{MV}$, the multiverse language, and $\mathcal{L}_\mathsf{\in}$, the language of set theory, as expressed by a `translation function', which, in turn, shows that $\mathcal{L}_\mathsf{MV}$ is a \textit{sublanguage} of $\mathcal{L}_\mathsf{\in}$. As a consequence of this, the existence of the core of $\mathsf{MV}$ has bearings on
set theory, as `standardly' construed as the theory of $V$, thus, Steel's Programme makes full sense. In broader philosophical terms, the interaction between the two `languages' may be construed as a way to provide set-theorists with a strategy to respond to the following question:

\begin{question}\label{mainquestion}

Is there a `preferred' universe of sets?

\end{question}

Clearly, if the Programme is successful, then the response to this question is in the positive, a fact which would have a considerable impact on our understanding of set theory, in particular, of its foundations.

The main purpose of the paper is that of providing a comprehensive assessment of the Programme, by focussing on three main topics: (1) its `evidential framework': in particular, the use of LCs and of forcing extensions of $V$ as `worlds' in the $\mathsf{MV}$ axiomatic set-up (section 2); (2) the core: the assumptions needed for its existence, and its level of (in)determinacy over $\mathsf{MV}$ (section 3 and 4); (3) the hypothesis that Ultimate-$L$ is the core, and the justification for this claim (section 5).

But first, we would like to introduce very briefly the general features of the wider philosophical context within which Steel's Programme may be discussed.

\subsection{Two Kinds of Universism}

$\mathsf{MV}$ locates itself in the current universe/multiverse debate, and, consequently, in the debate on \textit{pluralism}, that is, on the issue of whether mathematical truth splits into \textit{many}, mutually incompatible, truths.\footnote{For an overview of the debate, see \cite{koellnerxxx}, and \cite{afht2015}.} Pluralism is very often taken to correspond to `ontological' pluralism, that is, to the view that:

\vspace{11pt}

\noindent
\textbf{(Ontological) Pluralism}.  There are many, \textit{alternative} universes of set theory (there is a set-theoretic \textit{multiverse}).\footnote{This viewpoint is most prominently represented by \cite{hamkins2012}.}

\vspace{11pt}

\noindent
But some pluralists would rather commit themselves to semantic pluralism, that is, to the view that the \textit{truth-value} of all set-theoretic statements undecidable from the $\mathsf{ZFC}$ axioms is indeterminate (neither true nor false). The opposite camp is represented by:

\vspace{11pt}

\noindent
\textbf{(Ontological) Non-Pluralism}. There is just \textit{one} universe of set theory.\footnote{For arguments against set-theoretic pluralism, see \cite{martin2001} and \cite{koellnerxxx}; for the potential skeptic consequences of multiversism, see \cite{button-walsh2018}, pp. 206-213, and \cite{barton2016}.}

\vspace{11pt}

\noindent
In what follows, we will mostly refer to the position above as:

\vspace{11pt}

\noindent
\textbf{Classic Universism}. Set theory is the theory of a \textit{single} universe, $V$, whose features are pinned down by the $\mathsf{ZFC}$ axioms (and, potentially, by other axioms extending $\mathsf{ZFC}$).

\vspace{11pt}

\noindent
Steel himself introduces and considers several theses, of different nature and strength, concerning set-theoretic ontology and truth. The philosophical thesis which postulates the existence of the core of $\mathsf{MV}$ is what Steel calls Weak Absolutism:

\vspace{11pt}

\noindent
\textbf{Weak Absolutism}. The multiverse has a core (a `reference' or `preferred' universe).\footnote{\cite{steel2014}, p. 168. On Steel's own classification, our Classic Universism corresponds to Strong Absolutism, whereas Ontological Pluralism, presumably, corresponds to Strong Relativism (which, however, isn't described by Steel). Weak Relativism is the thesis that the whole of $\mathcal{L}_\mathsf{\in}$ is `meaningful' in $\mathcal{L}_\mathsf{MV}$, a position which leaves it open whether there really exists a `reference universe' in $\mathcal{L}_\mathsf{MV}$.}

\vspace{11pt}

\noindent
We may re-state Weak Absolutism in a slightly different, but maybe more perspicuous, way, as follows:

\vspace{11pt}

\noindent
\textbf{Core Universism}. Set theory is the theory of multiple (set-theoretic) universes, each of which contains a unique \textit{core universe}, which has a better claim to be seen as the `preferred' universe of sets than any other universe.

\vspace{11pt}

To state very quickly the main differences between the two positions: the Classic Universist may be standardly characterised as someone believing that our \textit{intuition} of sets, or the `concept of set' itself, will provide us with a \textit{unique}, consistent extension of the $\mathsf{ZFC}$ axioms which will \textit{uniquely} fix the truth-value of the undecidable statements.\footnote{For a classic articulation of this position, see \cite{godel1947}. For an overview of alternative conceptions justifying the axioms, see \cite{incurvati2020}.} By contrast, the Core Universist may be characterised as someone who takes all alternative `universes' as equally legitimate; however, such a Universist will also hold that each \textit{universe} (or, if you wish, \textit{theory}) contains `traces' of a single, `preferred' universe, and much of the value of the position consists in showing that the claim is true, that is, that a core universe is really \textit{detectable} within the multiverse itself. Of course, the Core Universist also expects to be able to describe the properties of the core in a satisfactory way.

In order to attain a reduction of set-theoretic incompleteness, the Classic Universist will suggest further exploration of our intuitions about sets, or sharpening of the concept of set, whereas the Core Universist will suggest further exploration of the properties of the core through the multiverse axioms.

Now, it is clear that Steel's Programme advocates the Core Universist's standpoint, and, therefore, has deep implications on the preferability of Core over Classic Universism: as already said, the Programme, if successful, would lend support to Core Universism. Indeed, the Core Universist's construal of the Programme's goals and results could be condensed as follows: `Non-pluralism about set-theoretic ontology cannot be correct, as we are aware of the existence of many alternative universes (as well as of alternative theories extending $\mathsf{ZFC}$). However, given a suitable version of the multiverse, one resting upon significant bits of current set-theoretic practice (upon the `evidential framework' addressed in the next section), we may identify a `preferred' universe within the multiverse itself. But then, for all our foundational purposes, such a universe might be seen as a fully adequate incarnation of our pre-theoretic notion of `single universe', perhaps not exactly the Classic Universist's one, indeed, a more `pragmatic', but equally justified, version of it.'

The Weak Absolutist/Core Universist's position has already been vindicated, in part, by Usuba's result: if there are sufficiently strong LCs, then $\mathsf{MV}$ has a core. However, whether, and in what sense, the core could be seen as a `preferred universe', and, in the presence of the translation function from $\mathcal{L}_\mathsf{MV}$ to $\mathcal{L}_\mathsf{\in}$, as the equivalent of the Classic Universist's $V$, is still open to debate, as we shall see in the next sections.

\section{The Evidential Framework of $\mathsf{MV}$}

In the preliminaries of \cite{steel2014}, Steel introduces and advocates a few distinctive positions, which he takes to be the main motivation for, and underlying conceptual framework of, his multiverse conception, as incarnated by the $\mathsf{MV}$ axioms. We summarise them below:

\begin{enumerate}

\item Large Cardinals are `practically' \textit{necessary} to extend $\mathsf{ZFC}$, mostly as a consequence of the following two phenomena:

\begin{enumerate}

\item LCs `calibrate' the consistency of extensions of $\mathsf{ZFC}$, and of  \emph{most} undecidable set-theoretic statements;

\item LCs maximise \textit{interpretative power}, insofar as the hierarchy of consistency strengths of Large Cardinal Axioms (LCAs) is very aptly matched by the hierarchy of proof-theoretic strengths of extensions of $\mathsf{ZFC}$.\footnote{\label{determinacy}\cite{steel2014}, p. 162, also points out that additional confirmation of the `correctness' of LCs is provided by the multiple correspondences between these and Determinacy Axioms; in turn, the successfulness of Determinacy Axioms would be demonstrated by their usefulness and fecundity in many areas of mathematical research. For an overview of the connections between Determinacy and LCs, see \cite{maddy1988b} and \cite{koellner2014}.}

\end{enumerate}

\item Set theory should fundamentally be seen as the theory of the \emph{forcing extensions} and \emph{inner models} of models of the theory: $\mathsf{ZFC}+$LCAs. This position, in turn, rests upon the fact that independence proofs are practically carried out in the model theory of (fragments of) $\mathsf{ZFC}+$LCAs and, most crucially for Steel's purposes, also upon the fact that all `natural' theories extending $\mathsf{ZFC}$ may be mutually connected through using models with LCs.

\item Set theory, as currently practised and interpreted, splits into several `natural' theories, all of which extend $\mathsf{ZFC}$. Part of the rationale for adopting set theory as a foundation of mathematics, thus, consists in describing all such theories and their connections through exploiting, in turn, connections between LCs and models containing them.\footnote{\label{steel} The point is made by Steel in section 5 of \cite{steel2014}, p. 164. However, in private correspondence with the authors (26/10/2020), Steel has understated this fact as a motivation for $\mathsf{MV}$, and placed a lot more emphasis on the existence of the translation function from $\mathcal{L}_{\mathsf{MV}}$ to $\mathcal{L}_{\in}$ as the main rationale for formulating $\mathsf{MV}$.}

\item In order to describe all such theories, one should make use of \textit{forcing}. To be sure, models obtained through forcing have become an essential tool to produce a wide range of set-theoretic `universes' upholding or violating set-theoretic properties, and, in particular,  models obtained through forcing are used to prove the \textit{equiconsistency} of theories with LCs. So, in practice, \textit{forcing extensions} of the universe should be taken to stand for different universes (Steel's \textit{worlds}).\footnote{However, notice that forcing extensions of $V$ may produce violations of LCs. So, if one wants to preserve LCs uniformly, one should rule out several models obtained through different kinds of \textit{class forcing}. Cf. \cite{steel2014}, p. 167.}

\end{enumerate}

In the next subsections, we wish to discuss at length salient aspects of the points above and, through this, provide an assessment of the evidence for the $\mathsf{MV}$ axioms.

\subsection{Natural Theories, Large Cardinals, and Worlds}\label{2.1}

As stated in bullet point (3.) above, one of the purposes of Steel's multiverse conception is precisely that of `representing' all `natural' theories extending $\mathsf{ZFC}$ within a \textit{unified} axiomatic framework, without explicitly and directly having to deal with `universes'.

However, in order to show that the axioms are not \textit{semantically} vacuous, there will have to be objects fixing their interpretation, and such objects are, on the one hand, \textit{sets} and, on the other, \textit{worlds}\footnote{\cite{steel2014}, p. 165.}. But notice the peculiar use of worlds in $\mathsf{MV}$: the latter are introduced, and described, to account for the \textit{representability} of \textit{different} theories.\footnote{Both \cite{maddy2017}, pp. 310ff., and \cite{maddy-meadows2020}, pp. 7-8 stress the importance of this fact, by contrasting Steel's conception with those of Hamkins and Woodin, which would, on the contrary, openly commit to significantly more \textit{ontological} forms of multiversism, and to a `metaphysics of universes' from the beginning.}  The strategy is, crucially, facilitated by both proof-theoretic and model-theoretic facts concerning LCs, in particular, the following conjecture:

\begin{conjecture}\label{Con1}

Any `natural' extension of $\mathsf{ZFC}$ is either equiconsistent with $\mathsf{ZFC}$ or equiconsistent with $\mathsf{ZFC}+A$, where $A$ is a LCA. Moreover, the consistency strengths of `natural' extensions of $\mathsf{ZFC}$ are well-ordered.\footnote{The qualifier `natural' in the statement of Conjecture \ref{Con1} is necessary, for there are (contrived, hence `unnatural') examples of consistent sentences $\varphi$ whose consistency strength is strictly stronger than $\mathsf{ZFC}$ (meaning that $\mathsf{ZFC}$ does not prove that $Con(\mathsf{ZFC})$ implies $Con(\mathsf{ZFC} + \varphi)$), yet $\mathsf{ZFC} + \varphi$ does not even prove the consistency of $\mathsf{ZFC} + Con(\mathsf{ZFC})$. Hence the consistency of $\mathsf{ZFC} + \varphi$ does not even yield the consistency of $\mathsf{ZFC}+$`There exists an inaccessible cardinal'. }

\end{conjecture}

The main upshot of Conjecture \ref{Con1} is that, since all theories $\mathsf{ZFC}$+LCA considered so far are arranged in a well-ordered scale of consistency strengths,\footnote{Pending open questions about comparability or equivalence of the consistency of certain large-cardinal notions (e.g., the equiconsistency of \textit{supercompact} and \textit{strongly compact cardinals}), which presumably will eventually be resolved.} then also all natural theories are. Thus, the `invisible web' binding together all natural theories is, finally, cast into sharp relief by the proof-theoretic connections among all LCAs.

The next step consists in exploiting the fact that, using LCs, one is able to construct models, in particular \textit{set-forcing extensions} and \textit{inner models}, which satisfy any natural theory. Using this fact, and Conjecture \ref{Con1}, Steel is, therefore, able to formulate, in full, the main meta-theoretic constraint presiding over his `multiverse' (bullet point (2.) above):

\vspace{11pt}

\noindent
\textbf{Meta-Theoretic Constraint}. $\mathsf{MV}$'s worlds are just those models of $\mathsf{ZFC}$+LCA which are needed to link all natural theories extending $\mathsf{ZFC}$.

\vspace{11pt}

Finally, since inner models may well be defined inside forcing extensions, what we just need to characterise worlds are LCs and forcing: the $\mathsf{MV}$ axioms reflect this state of affairs, by quantifying over set-forcing extensions (and their grounds) with LCs.

We will deal with the models of $\mathsf{MV}$ later on, but, first, we wish to express several concerns about Steel's use of, and primary reliance upon, LCs, and about the notion of `naturalness'.

The first qualm refers to Conjecture \ref{Con1}, that all `natural' extensions of $\mathsf{ZFC}$ are equiconsistent with either $\mathsf{ZFC}$ or some LCA. While no evidence against it has been found thus far, the conjecture is far from being settled, and things are compounded by the fact that we lack a general definition of `large cardinal'. Now, it would certainly be unfair to have the \textit{unsharpness} of the notion count against Steel's project; however, the absence of ultimate evidence that any undecidable statement will turn out to be equiconsistent with a LCA is a fact, which casts some doubts on the tenability of the conjecture.

A second, possibly more malignant, issue is that the notion of `natural extension of $\mathsf{ZFC}$' is unclear. What we are told by Steel, at the very outset, is that a set-theoretic statement does qualify as `natural' if it is consistent with $\mathsf{ZFC}$, asserts some `facts' about sets, and is not of a metamathematical or proof-theoretic nature, but this is really not much.\footnote{Cf. \cite{steel2014}, p. 157: `By `natural' we mean considered by set theorists, because they had some set-theoretic idea behind them.'}  Now, turning things around, one could be tempted to extrapolate the notion of `naturalness' from the very context it is used for, namely, Conjecture \ref{Con1}; that is, one could define a set-theoretic statement $\varphi$ to be `natural' if $\mathsf{ZFC}$ plus some LCA proves that $\varphi$ holds in some definable (allowing for set parameters) inner model of $\mathsf{ZFC}$ of a forcing extension of $V$ (including definable, with set parameters, class forcing extensions that preserve $\mathsf{ZFC}$). By this interpretation, all $\mathsf{ZFC}$ axioms as well as all LCAs are natural, and so are CH, V=L, V=HOD, SCH, as well as their negations. In fact, all set-theoretic statements known to be consistent with $\mathsf{ZFC}$ and asserting some facts about sets are of this sort. But this move would also be problematic, for a different reason: `equiconsistency with LCs' does not square very well with the common-sense, intuitive meaning of `naturalness'.

Turning to the notion of `naturalness', a major concern, in this case, is that $\mathsf{MV}$'s scope for `naturalness' is too restrictive from the beginning, as it leaves out theories, such as $\mathsf{ZF}+$AD, which unquestionably express deep set-theoretic facts. Now, one could think that there is a special relationship between theories extending $\mathsf{ZFC}$ and LCs, that other theories do not enjoy, but this is provably false: theories without Choice are \textit{equiconsistent} with LCs, may incorporate strong set-theoretic statements (such as the Axiom of Determinacy),\footnote{See footnote \ref{determinacy}.} and may also be compatible with further very strong LCs (such as Reinhardt and Berkeley Cardinals).\footnote{\label{LCBC}For these recently investigated large cardinal notions, see \cite{bagaria-koellner-woodin2019}, sections 2-3, pp. 289-296. It should be noted that it is presently not known whether Choiceless Large Cardinals are consistent with $\mathsf{ZF}$. If consistent, these LCs would have a dramatic impact on other set-theoretic hypotheses, such as the Ultimate-$L$ Conjecture, and the HOD Dichotomy Hypothesis. In particular, they would imply that HOD is `far' from $V$, so, in a sense, they may also be taken to have noticeable \textit{maximising} virtues (so long as $V$=HOD is construed as a limiting hypothesis). Reinhardt cardinals may be seen as instantiating a different form of maximality, as they imply a `maximisation' of the level of resemblance of $V$ with itself in the non-trivial embedding $j: V \rightarrow V$ which is used for their definition. For further details and philosophical considerations on all these issues see, again, \cite{bagaria-koellner-woodin2019}, section 8, p. 309ff. } Therefore, clearly, `equiconsistency with LCs' is manifestly insufficient, as a criterion, to incorporate theories which are incompatible with Choice.

Of course, Steel's exclusive focus on `natural extensions of $\mathsf{ZFC}$' is deliberate; but, then, we are left with no indication whatsoever about why, in the light of the other bits of evidence for $\mathsf{MV}$, $\mathsf{ZFC}$ is preferable over other axiomatisations.

\subsection{Large Cardinals as Maximality Principles}

Steel's preference for LCs is also motivated by another principle, that he calls `maximise interpretative power', which might be seen as consisting of two parts:

\vspace{11pt}

\noindent
\textbf{Maximise Interpretative Power [MIP]}. (A.) The $\mathsf{MV}$ axioms should be able to `represent' as many theories (`natural extensions of $\mathsf{ZFC}$') as possible; (B.) all the theories represented by the $\mathsf{MV}$ axioms should be such that, for any two of them, $T$ and $S$, if $Con(T) \rightarrow Con(S)$, then $Th(S) \subseteq Th(T)$.\footnote{Cf. \cite{steel2014}, p. 165.}

\vspace{11pt}

\noindent
LCAs constitute a paradigmatic case study with respect to \textbf{MIP}. As far as (A.) is concerned, we have seen that each `natural' theory $T$ is \textit{satisfied} in a forcing extension or inner model of another `natural' theory $S$, provided both $T$ and $S$ are equiconsistent with LCAs; as regards (B.), we know that the amount of mutual \textit{interpretability} of natural theories rises in proportion with their consistency strength. Thus, in the end, it would be legitimate to expect that:

\vspace{11pt}

\noindent
(*) As natural theories proceed up the large cardinal hierarchy in consistency strength, they agree on an ever-increasing class of mathematical statements.\footnote{Cf. \cite{maddy-meadows2020}, p. 16, from which (*) has been verbatim reproduced.}

\vspace{11pt}

\noindent
A few observations are in order. The first one, concerning (B.), is that, so far, \textbf{MIP} has been shown to be satisfied by LCAs only partially, that is, for specific kinds of sentences in the L\'evy hierarchy of arithmetical sentences. The following is the most one can hope to prove so far:

\vspace{11pt}

\noindent
\textbf{Empirical Fact.}
For any two theories $T, S$ whose consistency strength is at least that of the theory: $\mathsf{ZFC}$+`there exist infinitely many Woodin cardinals', such that $Con(T) \rightarrow Con(S)$, we have that $Th_{\Pi^{1}_\omega}(S) \subseteq Th_{\Pi^{1}_\omega}(T)$.

\vspace{11pt}

As a consequence, the applicability of \textbf{MIP} to LCAs has only been verified up to the level of second-order arithmetic.

The second observation is as follows. Steel's purpose is that of \textit{representing} a multiplicity of theories, all of which extend $\mathsf{ZFC}$+LCs, and we wonder whether \textbf{MIP} is really compatible with this goal. For suppose (B.) were applicable to \textit{all} sentences in the arithmetical hierarchy; then, clearly, all the theories targeted by (A.), would not, just, be \textit{represented}, but, for all practical purposes, they would rather be \textit{amalgamated} into just one theory. One way out of this difficulty would be to see (A.) as being sanctioned by the presently limited range of applicability of (B.), but this wouldn't help fully ease the tension between (A.) and (B.).

Finally, although this is very speculative, the hierarchy of LCs contradicting Choice might, potentially, be more successful at instantiating \textbf{MIP} than the hierarchy of LCs with Choice, but, as we have seen, none of these theories features among those targeted by $\mathsf{MV}$.\footnote{For other forms of maximality Choiceless LCs may potentially embody, see the considerations in footnote \ref{LCBC}.}

\subsection{Models for $\mathsf{MV}$}\label{MV}

A major asset of $\mathsf{MV}$ is that this theory is complete. We will not delve into the philosophical reasons for preferring a \textit{complete} axiomatisation of the multiverse over one which is not, as the task has already been carried out satisfactorily.\footnote{\label{mvcomplete}\cite{maddy-meadows2020}, pp. 20ff. The proof of the completeness of $\mathsf{MV}$ (Theorem 8) is on pp. 60-61.} In this subsection, we would rather like to focus on the semantics of $\mathsf{MV}$, in particular, on its `natural' models, $M^G$,\footnote{See further, paragraph below the enunciation of the $\mathsf{MV}$ axioms.}
and bring to light a slightly different mathematical approach to it (Lemma \ref{homo}).

We start with reviewing, very quickly, the axioms. The language of $\mathsf{MV}$ is the first-order language of set theory with two sorts, namely \emph{Set} and \emph{World}. We introduce a minor tweak to Steel's original formulation of the axioms. As we already know, $\mathsf{MV}$ has, as its own base, the axioms of $\mathsf{ZFC}$ plus LCs (we shall mostly refer to the \textit{base theory} as $T$). Now, let a $T$-theory be a theory extending $\mathsf{ZFC}$ which is preserved by set-forcing extensions, and by going to set-forcing grounds: it turns out that any theory of the form $\mathsf{ZFC}$+`there exists a proper class of some kind of LCs' is a $T$-theory. Then, the $\mathsf{MV}$ axioms for $T$ $(\mathsf{MV}_T)$, which we will be mostly referring to and using throughout the paper, are the following ones:

\begin{enumerate}
\item (Extensionality for Worlds) If two worlds have the same sets, then they are equal.
\item Every world is a model of $T$.
\item Every world is a transitive proper class. An object is a set if and only if it belongs to some world. All worlds have the same ordinals.
\item If $W$ is a world and ${\mathbb P} \in W$ is a poset, then there is a world of the form $W[G]$, where $G$ is ${\mathbb P}$-generic over $W$.
\item If $U$ is a world and $U=W[G]$, where $G$ is ${\mathbb P}$-generic over $W$, then $W$ is also a world.
\item (Amalgamation) If $U$ and $W$ are worlds, then there are posets ${\mathbb P}\in U$ and ${\mathbb Q}\in W$, and sets $G$ and $H$ ${\mathbb P}$-generic and ${\mathbb Q}$-generic over $U$ and $W$, respectively,  such that $U[G]=W[H]$.
\end{enumerate}

If $M$ is a countable model of $T$ and $G$ is $Coll(\omega, <Ord)^M$-generic over $M$, let $M^G$ be the model whose sets are those in $M[G]$ (in the case $M$ is ill-founded, then  $M[G]$ is defined accordingly) and whose worlds are the grounds of models of the form $M[G\restriction \alpha]$, for some $\alpha \in Ord^M$. It can be easily shown that $M^G$ is a model of $\mathsf{MV}_{T}$, when $T=\mathsf{ZFC}$.\footnote{\cite{maddy-meadows2020}, Theorem 26, p. 60.}  Moreover, if $T$ is obtained by adding to $\mathsf{ZFC}$ axioms such as `There is a proper class of $P$-cardinals', where $P$ stands for any of the usual large-cardinals properties, then it can also be proved that $M^G$ is a model of $\mathsf{MV}_{T}$.%

The collection of all models $M^G$ provides a complete semantics, and the axiom which guarantees the completeness of $\mathsf{MV}_T$ is, as has been shown by \cite{maddy-meadows2020}, p. 25, Amalgamation. But, as stressed by the authors, a consequence of this fact is that, in any model of the form $M^G$, not all generic filters for posets in $M$ may be taken into account to produce \textit{forcing extensions} which act, as required by the Meta-Theoretic Constraint of section \ref{2.1}, as the 'worlds' of $\mathsf{MV}$, but only those which are produced by the $Coll(\omega, Ord^M)$-generic filter $G$ over $M$.

We now proceed to prove the following:

\begin{lemma}
\label{homo}
The $\mathsf{MV}_T$ axioms imply that the multiverse is a homogeneous class-forcing  extension of each of its worlds.
\end{lemma}

\begin{proof} Since the model $M^G$ gives a complete semantics for $\mathsf{MV}_T$, we may assume that every model of $\mathsf{MV}_T$ is of this form. Now, working in a model $M^G$ as above, let $W$ be a world. So  $W$ is a ground of a model  of the form $M[G\restriction \alpha]$, for some $\alpha \in Ord^M$.

Let  ${\mathbb P}\in W$ be a poset such that for some $H_0$ ${\mathbb P}$-generic over $W$, $W[H_0]=M[G\restriction \alpha]$, and let $\kappa \geq \alpha$ be an uncountable $W[H_0]$-cardinal such that the cardinality of ${\mathbb P}$, as computed in $W$, is less than $\kappa$. Let $H_1$  be $Coll(\omega , <\kappa +1) /G\restriction \alpha$-generic over $M[G\restriction \alpha]$ so that $M[G\restriction \alpha][H_1]=M[G\restriction \kappa +1]$. Thus, $W[H_0][H_1]=M[G\restriction \kappa +1]$ is a generic extension of $W$ by a poset of cardinality $\kappa$ that collapses $\kappa$ to $\omega$, hence by Kripke's theorem (\cite{jech2003}, Lemma 26.7) equivalent to $Coll(\omega , \kappa)$.  Since $Coll(\omega , \kappa)$ is homogeneous, the lemma follows.
\end{proof}

\noindent
The following Corollary, based on the previous Lemma, is immediate:

\begin{corollary}
\label{coro1}
The $\mathsf{MV}_T$ axioms imply that if $W$ is a world and $G$ is ${\mathbb P}$-generic over $W$ for some poset ${\mathbb P} \in W$, then $W[G]$ is also a world.
\end{corollary}

\begin{proof}
Fix a world $W$ and a poset $\mathbb{P}\in W$. By Axiom 4, there is some $\mathbb{P}$-generic filter $G_0$ over $W$ such that $W[G_0]$ is a world. So, there exists $p\in \mathbb{P}$ that forces the statement `$W[\dot{G}]$ is a world', where $\dot{G}$ is the standard $\mathbb{P}$-name for the generic filter. Suppose, for a contradiction, that some condition $q\in \mathbb{P}$ forces the statement `$W[\dot{G}]$ is not a world'. By the lemma, let $\mathbb{Q}$ be a homogeneous collapse forcing in $W$ such that  $\mathbb{P}$ completely embeds into $\mathbb{Q}$, so we may as well assume that $\mathbb{P}$ is a complete suborder of $\mathbb{Q}$. Since the collapse forcing $\mathbb{Q}$ is homogeneous, there is an automorphism of $\mathbb{Q}$ sending $p$ to $q$ that fixes $\mathbb{P}$ (\cite{jech2003}, Lemma 26.9), which is impossible, as $p$ and $q$ force contradictory statements.
\end{proof}

\subsection{$M^{G}$ and The Translation Function}

As already anticipated in section 1.1, Steel's conception rests upon the crucial fact that the language of $\mathsf{MV}$, $\mathcal{L}_\mathsf{MV}$, may be seen as a sublanguage of $\mathcal{L}_\mathsf{\in}$. In order to show this, Steel defines a recursive translation function $t$ from $\mathcal{L}_{\mathsf{MV}}$ into $\mathcal{L}_{\in}$, such that:

\[\mathsf{MV}_T\vdash \varphi \quad \mbox{ if and only if } \quad T\vdash t(\varphi) \tag{Transl} \]

\noindent
where $T$, as said, is $\mathsf{ZFC}$+LCs. The translation function may  be rendered more transparently in terms of the semantics of $\mathsf{MV}$ as follows:

\begin{theorem}[Translation Function]\label{Transl}

For any sentence $\varphi$ of $\mathcal{L}_{\mathsf{MV}}$, every countable model $M$ of $\mathsf{ZFC}$ and every $G \ Coll(\omega, <Ord^M)$-generic over $M$

\[M^G\models \varphi \quad \mbox{ iff } \quad M\models t(\varphi) \tag{Transl}\footnote{\cite{steel2014}, p. 166. For further details on (Transl), see \cite{maddy-meadows2020}, section 5, pp. 30-38.}\]

\end{theorem}

\noindent
It should be noted that, in the presence of the specified semantics of $\mathsf{MV}$, the sentence $t(\varphi)$ asserts `$\varphi$ is true in all multiverses obtained from me'. Equivalently, `$\varphi$ is true in some multiverse obtained from me'. Now, as said (section 2.3), Amalgamation is needed to show that the models $M^G$ provide a complete semantics for $\mathsf{MV}_T$. It would, then, be natural to ask the following question:

\begin{question}

Is it possible to define a `translation function' in the context of multiverse axioms which do not imply Amalgamation?

\end{question}

\noindent
The answer is yes, and the proof is in Appendix B.

\section{The Core of $\mathsf{MV}$}

In this section, we will, on the one hand, show how the `core hypothesis' arises in the context of the $\mathsf{MV}$ axioms, in particular, how it is mathematically justified; this will help us provide an answer to the following question:

\begin{question}

Under what circumstances is the \textit{core} definable, and how?

\end{question}

On the other hand, we will show why Usuba's result (Theorem \ref{usuba} below) implies that $\mathsf{MV}_T$, where $T=\mathsf{ZFC}$+`there exists a proper class of extendible cardinals', proves that there exists a \textit{core universe}.

However, before proceeding to review all such results, we address the `core hypothesis' in the context of \textit{just} $\mathsf{ZFC}$, and recall relevant set-theoretic geological concepts and theorems which make sense of the hypothesis, and which will be instrumental to present our own results in section 4.

\subsection{Prelude: The (Outer) Core of $\mathsf{ZFC}$}

Initiated by Reitz and Hamkins a few years ago, set-theoretic geology has brought forward a very innovative approach to `universes of set theory' producible using forcing.\footnote{Fundamental references here are \cite{reitz2007}, and \cite{hamkins-reitz-woodin2008}. A comprehensive account of the programme's conception and results is in \cite{fuchs-hamkins-reitz2015}, but see also \cite{hamkins2012}, pp. 443-47, for further philosophical discussion.} Here follows a very brief review of its fundamental concepts.

Take $V$ to be a model of the axioms $\mathsf{ZFC}$. Hypothetically, it might be that $V$ is a forcing extension of a ground model $W$, that is, that there exists a $W$-generic filter $G \subseteq \mathbb{P} \in W$ such that $V=W[G]$. If this is the case, then it makes sense to explore the
`geology' of $V$, that is, the collection of grounds, the generic extensions of grounds and their grounds, and so on, which $V$ might contain, where a ground of a model $M$ is a model $N$ of $\mathsf{ZFC}$ such that $M$ has been obtained through forcing over $N$, that is, as an extension $M=N[G]$, where $G$ is a generic filter of a partial order $\mathbb{P} \in N$. Further `geological' notions will then crop up, which can be spelt out as follows.

The \textit{mantle} $\mathbb{M}$ is the intersection of all the \textit{grounds} of a model $M$ of $\mathsf{ZFC}$, and the \textit{bedrock} is a class $W$ which is a \textit{minimal} ground of $V$. By results of Usuba \cite{U:DDG}, the mantle is a model of $\mathsf{ZFC}$. Since \cite{reitz2007}, the following axiom has proved to be central to all geological investigations:

\begin{axiom}[Ground Axiom (GA)]

$V$ is not obtained from any non-trivial set-forcing over any ground.

\end{axiom}

Now, if the mantle itself satisfies GA, then the mantle is a \textit{bedrock}, in particular, a \textit{minimum} bedrock contained in all other grounds, and the latter could legitimately be seen as a \textit{core universe} of the set-generic multiverse generated over it.\footnote{\label{DDG}An essential property to be satisfied for this to take place is Downwards Directedness, that is, that any two grounds have a ground \textit{in common}.  The Downwards Directedness Hypothesis (DDG) is, precisely, the claim that all grounds are \textit{downwards directed}. \cite{U:DDG} proves that DDG already holds in $\mathsf{ZFC}$ (Proposition 5.1, p. 13).} A fuller mathematical characterisation of the core may, thus, be attempted. However, already at this point, it emerges that the features of the core are (over $\mathsf{ZFC}$) awfully indeterminate. This is shown by the following, fundamental theorem:

\begin{theorem}[Fuchs, Hamkins, Reitz]\label{mantle}

Every model of $\mathsf{ZFC}$ can be the mantle of another model of $\mathsf{ZFC}$.\footnote{\cite{fuchs-hamkins-reitz2015}, p. 464, Main Theorem 4.}

\end{theorem}

In an attempt to both attain a more determinate core, and further investigate its nature, geology theorists have taken into account a different hypothesis, which can be expounded as follows. If the mantle does not satisfy GA, then it makes sense to take into account the `mantle of the mantle',
$\mathbb{M}^1=\mathbb{M}^{\mathbb{M}}$, and then, always under the assumption that GA is not satisfied, the `mantle of the mantle of the mantle' and so on. In other terms, it makes sense to take into account the iteration of the `mantle operation'. Now, iterating the mantle is not a trivial task, as there are technical aspects involved which may prevent one from even \textit{defining} the \textit{n}th iterates of the mantle.\footnote{See \cite{fuchs-hamkins-reitz2015}, p. 496ff.} Assuming these difficulties may be overcome, one might ask whether, for an $\alpha \in Ord$ iterate of the mantle, one has that $\mathbb{M}^{\alpha}=\mathbb{M}^{\beta}$, for all $\beta>\alpha$, that is, whether there is a \textit{minimal} $\alpha$ such that $\mathbb{M}^{\alpha} \models GA$. If there is one, then $\mathbb{M}^{\alpha}$ is said to be the \textit{outer core} of the initial model (of $V$, if $V$ was such model). However, in this case, as in the previous one where $\mathbb{M} \models GA$, Theorem \ref{mantle} will imply that also the outer core does not have \textit{determinate} features, namely, that it might satisfy a wide range of mutually incompatible properties.

Moreover, it has been shown that the outer core cannot be uniquely pinned down in the iteration process, as a recent result by Reitz and Williams has confirmed \cite{fuchs-hamkins-reitz2015}'s Conjecture 74,\footnote{Cf. \cite{reitz-williams2019}, p. 6, Theorem 3.1 and ff. Conjecture 74 is in \cite{fuchs-hamkins-reitz2015}, p. 497.} and shown that, for any $\alpha \in Ord$, the outer core of a model $M$ could be any $\mathbb{M}^{\alpha}$, that is, any $\alpha$-th iterate of the mantle of $M$ (possibly also including including $\mathbb{M}^{Ord}$, although no proof is currently available for this specific case).\footnote{\cite{fuchs-hamkins-reitz2015} even envisages a possible extension, over the theory $\mathsf{GBC}$, of the iteration of the mantle operation beyond $\mathbb{M}^{Ord}$ for some model $M$, something which would imply that $M$ has \textit{no} outer core at all. See \cite{fuchs-hamkins-reitz2015}, pp. 497-498.}

To sum up, $\mathsf{ZFC}$ alone does not itself guarantee the existence of a `core universe' and, moreover, over $\mathsf{ZFC}$, no full-fledged, definite mathematical characterisation of the core may arise. As we shall see, a slightly different scenario comes to light, if one takes onboard LCs.

\subsection{The Proof of the Existence}

\cite{steel2014} credits Woodin for observing that `if the multiverse has a definable world, then it has a unique definable world, and this world is included in all the others'.\footnote{\cite{steel2014}, p. 168.} We start this subsection by giving a  proof of this fact.\footnote{We thank John Steel for providing us, in private correspondence, with some details for the proof of Theorem \ref{defworld}.}

\begin{theorem}[Definability of the Core]
\label{defworld}
If the multiverse has a definable world, then it has a unique definable world. More precisely, suppose $\varphi$ and $\psi$ are formulas in the language of the multiverse with only one free variable for sets. Then,
$$\mathsf{MV}_T\vdash '\forall U,W (\forall x ((x\in U \leftrightarrow \varphi(x)) \wedge (x\in W \leftrightarrow \psi(x))\to U=W).'$$
\end{theorem}

\begin{proof}
Working in a model $M^G$ of $\mathsf{MV}_T$ (see section \ref{MV}), let $U$ and $W$ be worlds defined by $\varphi$ and $\psi$, respectively. Since $U$ and $W$ are transitive and contain the same ordinals, it will be sufficient to  show that $U\subseteq W$ by showing $V_\alpha^U\subseteq W$ by induction on $\alpha$. This is clear for $\alpha =0$, and also clear for $\alpha$ a limit ordinal provided it holds for all ordinals less than $\alpha$.
So suppose $V_\alpha^U \subseteq W$ and let us show $V_{\alpha+1}^U\subseteq W$. Note that $V_\alpha^U\subseteq W$ implies $V_\alpha^U \in W$. This is because
$$x \in V_\alpha^U \,\,\mbox{  iff  } \,\,W \models t('rank(v)\!<\!\alpha \mbox{  in   the unique world defined by } \varphi')[x].$$
Hence, $V_\alpha^U$ is definable in $W$ with parameter $\alpha$.

Now let $Y\subseteq V_{\alpha}^U$, $Y\in U$.  We must check that $Y\in W$. As in Lemma \ref{homo}, we can find
$\gamma$ large enough so that
$$W^{Coll(\omega,\gamma)}\models \mbox{'I am a forcing extension of $X$, for some world $X$ which }$$ $$\mbox{\hspace{1.8cm}is definable by }  \varphi  \mbox{  in the multiverse generated by me}.'$$
It follows that $U\subset W[H]$ for all $H$  $Coll(\omega,\gamma)$-generic over $W$. So
$Y\in W[H]$ for all such $H$. Since $Y\subseteq W$ and $Coll(\omega ,\gamma)$ is homogeneous, it follows that $Y\in W$.
\end{proof}

It follows that if there is a definable world $U$ in the multiverse, then it is unique and is contained in all of $\mathsf{MV}$'s worlds.\footnote{\cite{steel2014}, p. 168ff.} Clearly, since the core, if it exists, is definable, the core exists if and only if there is a definable world.

\subsection{The Core is the Mantle of $V$}

We proceed to introduce the fundamental result, due to Toshimichi Usuba, which shows that, under the assumption of sufficiently strong LCs, $V$ has a \textit{smallest} ground.

\begin{theorem}[\cite{U:ECM}, p. 72]\label{usuba}

Suppose there exists an extendible cardinal. Then the mantle is a ground of $V$. In fact if $\kappa$ is extendible, the $\kappa$-mantle of $V$ is its smallest ground.\footnote{For a cardinal $\kappa$, the $\kappa$-mantle is the intersection of all $\kappa$-grounds, that is of all grounds $W$ of $V$, such that there exists a $\mathbb{P} \in W$ of size less than $\kappa$, and generic $G \subseteq \mathbb{P}$, such that $V=W[G]$.}

\end{theorem}

\noindent
Now, since \textit{all} grounds are \textit{downwards-directed},\footnote{See footnote \ref{DDG}.} by using Theorem \ref{usuba}, we can now prove the following:

\begin{prop}[Existence of the Core of $\mathsf{MV}_T$]\label{coreexists}
Let $T$ be $\mathsf{ZFC}$+`there exists a proper class of extendible cardinals'. Then the $\mathsf{MV}_T$ axioms imply that the multiverse has a core, which is the mantle (and a ground) of every world in the multiverse.\footnote{Note the assumption, in Theorem \ref{coreexists}, that there exists a \textit{proper class}, not just \textit{one} extendible in all worlds of $\mathsf{MV}$. The reason for that is that a proper class of extendibles is needed, if one wants to preserve \textit{extendibles} in the set-forcing extensions of $V$ which count as worlds.}
\end{prop}

\begin{proof}
By Theorem \ref{usuba}, the mantle $\mathcal{M}_W$ of every world $W$ of the multiverse is a ground, hence by Axiom 5 of $\mathsf{MV}$, it is also a world. Now suppose $U_0$ and $U_1$ are worlds. By Amalgamation, they are also grounds of some world $W$ containing them. Hence, since the grounds are downwards-directed, $\mathcal{M}_{U_0}=\mathcal{M}_{U_1}$. It follows that all the worlds of the multiverse have the same mantle $\mathcal{M}$, which is the intersection of all the worlds. Thus, $\mathcal{M}$ is definable by the formula
$$\forall U (U\mbox{ is a world}\to x\in U).$$
Hence by Theorem \ref{defworld} the mantle $\mathcal{M}$ is the core of the multiverse.
\end{proof}

\subsection{The Strength of the Large-Cardinal Assumptions}

By Proposition \ref{coreexists}, if $M$ satisfies that there exists an extendible cardinal, then $M^G$ has a \emph{core}. It is legitimate to ask whether the large-cardinal assumption in Theorem \ref{usuba} might be weakened to the level of a LC lower in the consistency strength hierarchy; more generally:

\begin{question}\label{supercompact}

For what choices of $T$ does $\mathsf{MV}_T$ prove the existence of a core?

\end{question}

The following theorem shows that the existence of a proper class of supercompact cardinals is not sufficient to prove the existence of the core, thus suggesting a potential threshold for the choice of $T$ mentioned by Question \ref{supercompact}.

\begin{theorem}\label{threshold}
Assume there is a proper class of supercompact cardinals. Then there is a class forcing notion which forces that there exists a proper class of supercompact cardinals and  there is no core.
\end{theorem}

\begin{proof}
First, force with a class-forcing iteration with Easton support in order to make every supercompact $\kappa$  indestructible by $<\!\!\kappa$-directed-closed forcing (see \cite{U:AA}), followed by  the Jensen's class-forcing iteration that forces the GCH. Standard arguments show that all the supercompact cardinals are preserved and no new supercompact cardinals are created. Over the generic extension, make again all supercompact cardinals $\kappa$ indestructible by $<\kappa$-directed-closed forcing (which preserves the GCH and does not create new supercompact cardinals) and call the resulting model $V[G]$. Now force with a class-forcing iteration $\mathbb{P}$ with Easton support that forces a version of the  Continuous Coding Axiom (CCA), similarly as in \cite{reitz2007}, which codes every set of ordinals  proper-class-many times into the power-set function on the class $S$ of supercompact cardinals that are not limit of supercompact cardinals. Again, standard arguments show that all supercompact cardinals in $S$ are preserved and no new supercompact cardinals are created. Let $V[G][H_0]$ be this model. Finally, force over $V[G][H_0]$ with the class-forcing product $$\mathbb{Q}=\prod_{\kappa \in ORD}\mathbb{Q}(\kappa)$$
where $\mathbb{Q}(\kappa)$ is the forcing for adding a Cohen subset of $\kappa$, if $\kappa$ is in $S$, and is the trivial forcing otherwise. Let the resulting  model be $V[G][H_0][H_1]$. We claim  that supercompact cardinals in $S$ are preserved. So  let $\kappa\in S$.
Working in $V[G][H_0]$, note first that $\mathbb{Q}$ factors as $\mathbb{Q}_{\kappa} \times \mathbb{Q}_{[\kappa +1 , Ord)}$.
Also note that, since $\kappa$ is not a limit of supercompact cardinals,  ${\rm{sup}}(S\cap \kappa)<\kappa$. So, $\mathbb{Q}_{\kappa}$ factors as $\mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}\times \mathbb{Q}(\kappa)$, where $\mathbb{Q}(\kappa)$ is the forcing that adds a Cohen subset of $\kappa$. The product  $\mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}\times \mathbb{Q}(\kappa)$ is equivalent, as a forcing notion, to $\mathbb{Q}(\kappa)\times \mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}$. Also, since $\mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}$ has cardinality less than $\kappa$ and $\mathbb{Q}(\kappa)$ does not add new bounded subsets of $\kappa$, $\mathbb{Q}(\kappa)\times \mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}$ is equivalent to $\mathbb{Q}(\kappa)\ast \mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}$. Now, since $\mathbb{Q}(\kappa)$ is $<\kappa$-directed closed it preserves the supercompactness of $\kappa$, and then so does $\mathbb{Q}_{{\rm{sup}}(S\cap \kappa)}$ subsequently, since it has cardinality less than $\kappa$.   Moreover, $\mathbb{Q}_\kappa$ forces that $\mathbb{Q}_{[\kappa +1 , Ord)}$ is $<\kappa$-directed-closed, and therefore it also preserves the supercompactness of $\kappa$. Since $\kappa^{<\kappa}=\kappa$ for every $\kappa \in S$, the forcing $\mathbb{Q}$ preserves cardinals and the power-set function. Now we may argue similarly as in \cite{reitz2007} to show that every ground of $V[G][H_0][H_1]$ has a proper subground, hence there is no core.\footnote{\cite{reitz2007}, Theorem 3.9, p. 1309.} For suppose $W$ is a ground of $V[G][H_0][H_1]$. Let $\mathbb{P}$  be a poset in $W$ and $g$ a $\mathbb{P}$-generic filter over $W$ such that $V[G][H_0][H_1]=W[g]$. Since the two models $V[G][H_0][H_1]$ and $W$ agree on the values of the power-set function for a tail of elements of $S$, as well as on statements of the form
``$\kappa$ is the $\alpha$-th element of $S$" for a tail of cardinals $\kappa$ in $S$, every element of $V[G][H_0]$ is coded into the power-set function of $W$ on $S$, hence $V[G][H_0]\subseteq W$. Now let $\kappa$ in $S$ be greater than $|\mathbb{P}|$, and let $h_\kappa$ be the Cohen generic subset of $\kappa$ added by $H$. Since $V[G][H_0]\subseteq W$, every condition of the forcing $\mathbb{Q}(\kappa)$, therefore every bounded subset of $h_\kappa$, belongs to $W$. Hence, since the pair $\langle W, W[h]\rangle$ satisfies the $\kappa$-approximation property, $h_\kappa \in W$. Thus, writing $H_1$ as the product $H_1^{\leq \kappa}\times H_1^{>\kappa}$, we have that $V[G][H_0][H^{>\kappa}]\subseteq W$, and $V[G][H_0][H^{>\kappa}]$ is a ground of $W$. Now, if $\kappa_0 <\kappa_1$ are the first two members of $S$ greater than $|\mathbb{P}|$, then $V[G][H_0][H^{>\kappa_1}_1]$  is a proper ground of $V[G][H_1][H_1^{>\kappa_0}]$, hance also a proper ground of $W$.
\end{proof}

Therefore, although no formal evidence may currently be provided, it is reasonable to conjecture, that, as a consequence of Theorem \ref{threshold}, no LCA of consistency strength lower than that of `there exists an extendible cardinal' will be sufficient to prove the existence of the core. However, more work is needed to further enlighten the issue.

\section{The Core: CH and Forcing Axioms}

\subsection{The Core and CH}

We know that the core of $\mathsf{MV}$, under certain assumptions, exists, so it makes sense to find out what features it has and, based on these, what it may reveal to us about such \textit{undecidable} statements as CH. In particular, we would like to address the following question:

\begin{question}\label{questCH}
Does $\mathsf{MV}_T$ prove that the core satisfies the CH, where $T=\mathsf{ZFC}+$`there exists a class of extendible cardinals'?
\end{question}

The question may, in fact, extend to any other statement $\varphi$ which is not decided by $\mathsf{ZFC}$. In particular, for each such $\varphi$, one may ask whether the core implies the truth or falsity of $\varphi$.

The results which follow provide answers to such questions, and, overall, starkly expose the \textit{indeterminacy} of the core. We, first, outline the mathematical strategy for CH.

If the core exists, then it satisfies GA, namely, it does not have any proper grounds. \cite{reitz2007} proved that every model of $\mathsf{ZFC}$ has a class-forcing extension which preserves any desired $V_\alpha$, satisfies V=HOD, and is a model of $\mathsf{ZFC}$+GA.\footnote{\cite{reitz2007}, Theorems 3.5, and 3.10, pp. 1308-1310.} By results in \cite{bagaria-poveda2020}, the class-forcing used to obtain the model preserves extendible cardinals. So, starting with a model $M$ satisfying $T=\mathsf{ZFC}+$`There is a proper class of extendible cardinals' we may class-force over it to obtain a model $M[H]$ of $T$ satisfying GA and, e.g., $\neg$CH. And then by forcing with $Coll(\omega, <Ord)^{M[H]}$ over $M[H]$ we obtain a model of $\mathsf{MV}_T$ whose core is $M[H]$ and satisfies $\neg$CH.

Now we show how this strategy can be extended to all $\Sigma_2$ set-forceable statements $\varphi$:

\begin{theorem}\label{sigma2}
Let $\varphi$ be a $\Sigma_2$-statement (with parameters) that can be forced by set forcing. Assume there is a proper class of extendible cardinals. Then in some class-forcing extension of $V$ that preserves extendible cardinals the statement $\varphi$ holds in the core of the $\mathsf{MV}$ multiverse built around the extension itself.
\end{theorem}

\begin{proof}
First force $\varphi$ by set forcing. In the forcing extension let $\kappa \in C^{(1)}$ be such that $V_\kappa \models \varphi$. Then force the GCH above $\kappa$, in the usual way using class forcing, so that $V_\kappa$ is not changed. As in \cite{reitz2007}, we may further class-force to get a model of GA, so that $V_\kappa$ is still unchanged. By results contained in \cite{tsaprounis2014} and  \cite{bagaria-poveda2020}, both class-forcing notions  preserve extendible cardinals.\footnote{\cite{tsaprounis2014}, Lemma 4.3, p. 113 and \cite{bagaria-poveda2020}, Theorem 5.3, p. 16, and Theorem 7.5, p. 23.} Hence, since the final extension $M$ satisfies the GA, if $G$ is $Coll(\omega , <Ord)^M$-generic over $M$, then  the core of the $\mathsf{MV}$ multiverse given by $M^G$, which exists because in $M$  there exists a proper class of extendible cardinals, is $M$ itself. Also, since $V_\kappa$ has not been changed and satisfies $\varphi$, in $M$ we have that $V_\kappa \models \varphi$ and, moreover, $\kappa \in C^{(1)}$, because being in $C^{(1)}$ is characterized by being uncountable and satisfying $V_\kappa =H_\kappa$. Hence, $M\models \varphi$.
\end{proof}

\begin{corollary}
If $T=\mathsf{ZFC}+$`$\mbox{There exists a proper class of extendible cardinals}$' is consistent, then so is $T$ plus that the core of $\mathsf{MV}_T$ satisfies the CH, or $\neg$CH, with any possible value of the size of the continuum.
\end{corollary}

\begin{corollary}
If $T$ as above is consistent, then so is $T$ plus that the core of $\mathsf{MV}_T$ satisfies $\square_{\omega_2}$, or its negation.
\end{corollary}

\begin{proof}
$\square_{\omega_2}$ is a forcible, by countably-directed and $\omega_1$-strategically closed forcing, $\Sigma_1$ statement, with $\omega_1$ and $\omega_2$ as parameters. Also, its negation is forced by $Col(\omega_1, <\kappa)$, with $\kappa$ a Mahlo cardinal.
\end{proof}

The well-known forcing axiom $\mathrm{MA}_{\aleph_1}$ is also a $\Sigma_2$ statement, with $\aleph_1$ as a parameter. Thus,  Theorem \ref{sigma2} yields the following:

\begin{corollary}\label{MAaleph1}
If $T$ as above is consistent, then so is $T$ plus that the core of $\mathsf{MV}_T$ satisfies $\mathrm{MA}_{\aleph_1}$.
\end{corollary}

However, Theorem \ref{sigma2} does not hold   for  \textit{all} $\Pi_2$ statements. For instance, the fact that the mantle is not a ground can be expressed by saying that every ground has a further smaller ground, which is a $\Pi_2$ statement, without parameters. This is a class-forceable fact,\footnote{\cite{reitz2007}, Theorem 5.5, p. 1314.} but it implies the non-existence of the core.

\medskip

\subsection{The Core and Strong Forcing Axioms}

Corollary \ref{MAaleph1} helps us introduce, with fuller generality, the examination of the behaviour of the core with respect to forcing axioms. Now, what has neatly come to surface is that the core of $\mathsf{MV}_T$, where $T$ may, and even may not, have extendible cardinals, is also consistent with \textit{strong} forcing axioms. Corollary 3.8 of \cite{reitz2007} already proves that the Proper Forcing Axiom (PFA) is consistent with the GA.\footnote{\cite{reitz2007}, p. 1308.} We extend this result also to MM, MM$^{++}$, etc.

Recall that MM$^{++}$ states that for every poset ${\mathbb P}$ that preserves stationary subsets of $\omega_1$, every collection $\{ D_\alpha :\alpha <\omega_1\}$ of dense open subsets of ${\mathbb P}$, and every collection $\{ \sigma_\alpha :\alpha <\omega_1\}$ of ${\mathbb P}$-names for stationary subsets of $\omega_1$, there exists a filter $G\subseteq {\mathbb P}$ that is generic for $\{ D_\alpha :\alpha <\omega_1\}$ (i.e., $G\cap D_\alpha \ne \emptyset$, all $\alpha$), and such that $\sigma_\alpha [G]$ is a stationary subset of $\omega_1$, all $\alpha$. It is well-known that MM$^{++}$ can be forced, assuming the existence of a supercompact cardinal.

\begin{theorem}
If $\mathsf{ZFC}$ plus the existence of a supercompact cardinal is consistent, then so is MM$^{++}$ plus GA.
\end{theorem}

\begin{proof}
Let $V$ satisfy $\mathsf{ZFC}$ plus the existence of a supercompact cardinal. Force over $V$ to obtain a model of $\mathsf{ZFC}$ plus MM$^{++}$. Call this model $V[G]$. Then force with an $\omega_2$-directed-closed $ORD$-length iteration ${\mathbb P}$ over $V[G]$ to obtain a model $V[G][H]$ of GA while preserving cardinals (as in \cite{reitz2007}).  We claim that MM$^{++}$ holds in $V[G][H]$. For suppose  $\tau$ is a ${\mathbb P}$-name for a poset that preserves stationary subsets of $\omega_1$,  $\{ \dot{D}_\alpha : \alpha <\omega_1 \}$ is a ${\mathbb P}$-name for a collection of dense open subsets of $\tau$, and $\{ \dot{\sigma}_\alpha :\alpha <\omega_1\}$ is a ${\mathbb P}$-name for a collection of $\tau$-names for stationary subsets of $\omega_1$. Let $\lambda$ be a large enough cardinal such that $\tau$ and $\langle \dot{D}_\alpha : \alpha <\omega_1 \rangle$ and $\{ \dot{\sigma}_\alpha :\alpha <\omega_1\}$ are ${\mathbb P}_\lambda$-names. Since ${\mathbb P}_\lambda$ is $\omega_2$-directed-closed, arguing similarly as in \cite{larson2000}, we can show that ${\mathbb P}_\lambda$ preserves MM$^{++}$.

We claim that in $V[G][H_\lambda]$, the poset $\tau[H_\lambda]$ preserves stationary subsets of $\omega_1$: For suppose $S\subseteq\omega_1$ is stationary and $\dot{C}$ is a $\tau[H_\lambda]$-name for a club subset of $\omega_1$. Since the remaining part of the iteration does not add any new subsets of $\omega_1$, $S$ is also stationary in $V[G][H]$. Moreover, since being a club is absolute for transitive models, and every $\tau[H_\lambda]$-generic filter over $V[G][H]$ is also $\tau[H_\lambda]$-generic over $V[G][H_\lambda]$, we have that in $V[G][H]$, $\dot{C}$ is a $\tau[H_\lambda]$-name for a club subset of $\omega_1$. Hence, since in $V[G][H]$ the poset $\tau[H_\lambda]$ preserves stationary subsets of $\omega_1$, we have that $\Vdash_{\tau[H]}``\check{S}\cap \dot{C}\ne \emptyset"$. But the latter is absolute for transitive models, and so it holds in $V[G][H_\lambda]$.

Moreover, in $V[G][H_\lambda]$, $\{\dot{\sigma}_\alpha[H_\lambda] :\alpha <\omega_1\}$ is a collection of $\tau$-names for stationary subsets of $\omega_1$. Since MM$^{++}$ holds in $V[G][H_\lambda]$, there exists a filter $F\subseteq \tau[H_\lambda]$ that is generic for the collection $\{ \dot{D}_\alpha [H_\lambda]:\alpha <\omega_1\}$, and such that $\dot{\sigma}_\alpha[H_\lambda][F]$ is stationary. By absoluteness this is also true in $V[G][H]$. This shows that MM$^{++}$ holds in $V[G][H]$.
\end{proof}

It follows from the theorem above that if $T=\mathsf{ZFC}$+`there exists a supercompact cardinal' is consistent, then so is that the core of $\mathsf{MV}_T$ satisfies MM$^{++}$. Also, since the class-iteration ${\mathbb P}$ that forces the GA preserves extendible cardinals,\footnote{See, again, \cite{bagaria-poveda2020}, Lemma 7.4 and Theorem 7.5.} if the theory $T=\mathsf{ZFC}+$`there exists a proper class of extendible cardinals' is consistent, then so is $\mathsf{MV}_T$ plus that the core satisfies MM$^{++}$. Moreover, in both cases, by results in \cite{aspero-schindler2019}, the core also satisfies Woodin's $(\ast)$ axiom.

\section{Probing Steel's Programme: The Core as Ultimate-$L$}

Let's take stock. All the results illustrated in sections 3 and 4 show that, assuming a proper class of extendible cardinals, the core of $\mathsf{MV}$ exists, but is still a highly \textit{indeterminate} object. In particular, we have seen that the core may satisfy any of the strongest known forcing axioms, all of which imply that the continuum has size $\aleph_2$, and it may also satisfy CH, and, for that matter, any other $\Sigma_2$ set-forceable statement, with parameters  (Theorem \ref{sigma2}).

Two immediate considerations are in order. The first is that, based on the provable, under certain assumptions, existence of the core of $\mathsf{MV}$, the Universist may now feel legitimised to shift to the Core Universist view that $V$ \textit{is} the core, by settling on a theory, in $\mathcal{L}_{\in}$, which states this explicitly, that is, a theory which contains $V=\mathcal{C}$ as an axiom (henceforth, $\mathcal{C}$ will be our designated symbol for the core); in the next section we will examine this possibility in more detail.\footnote{See also a  discussion of this in \cite{maddy-meadows2020}, p. 42.} The second one is that, as a consequence of the persistent indeterminacy of the core also in the presence of LCs, even if the Core Universist decided to settle on $V=\mathcal{C}$ as the correct extension of $\mathsf{ZFC}$+LCs, she wouldn't still be able to fix the features of the core, and thus, the meaning, in $\mathcal{L}_{\in}$, of the undecidable statements.

So, it is time for Steel's Programme to kick in, and, in the next subsections, we will examine its execution and assess its prospects.

\subsection{Ultimate-$L$ and $\mathsf{MV}$}

In recent years, as a consequence of the far-reaching developments of the \textit{inner model programme}, the possibility of the existence of a canonical inner model for a  \textit{supercompact} cardinal has emerged. Woodin has called such a model Ultimate-$L$, insofar as this model would incorporate all features of an $L$-like inner model, as well as \textit{all} LCs, thus leading to the completion of the inner model programme itself.\footnote{See footnote \ref{Woodin}. For an accessible overview of the inner model programme, see \cite{jensen1995}; for technical results, \cite{devlin1984}, or \cite{kanamori2009}.} If such a model exists, then one could say that it represents an `optimal' approximation of $V$, something which would justify viewing the axiom $V$=Ultimate-$L$ as the most natural extension of $\mathsf{ZFC}$, and \cite{steel2014} has precisely taken into account such a possibility.\footnote{Although, in that work, Woodin's Ultimate-$L$ Conjecture is not directly addressed. See further, subsection 5.2.}

In this subsection, we will first be concerned with mathematical results about Ultimate-$L$ \textit{and} the core and, in the next one, we will articulate the Core Universist position based on these results. We start with  Woodin's definition of the axiom $V$=Ultimate-$L$.

\begin{definition}[$V$=Ultimate-$L$]
The axiom $V$=Ultimate-$L$ asserts:
\begin{enumerate}
    \item  There is a proper class of Woodin cardinals, \emph{and}

    \item For each $\Sigma_2$ sentence $\varphi$, if $\varphi$ holds in $V$, then there is a universally-Baire set $A\subseteq \mathbb{R}$ such that
    $$HOD^{L(A,\mathbb{R})}\models \varphi.$$
\end{enumerate}
\end{definition}

\noindent
Woodin has shown that $V$=Ultimate-$L$ implies the CH.\footnote{See \cite{woodin2017}, Introduction.} It also implies that $V$ has no non-trivial grounds, i.e., that $V$ is not a set-generic extension of any inner model, and that $V$=HOD.

As mentioned, it is an open question whether $\mathsf{ZFC}+$the existence of LCs (e.g., a proper class of Woodin cardinals, or even a proper class of supercompact, or extendible cardinals, etc.) implies the existence of an inner model satisfying $V$=Ultimate-$L$. However, Woodin has made the following conjecture. Recall that an inner model $N$ is a \emph{weak extender model for the supercompactness of $\delta$}  if for every $\lambda \geq \delta$ there is a normal fine measure $\mathcal{U}$ on $\mathcal{P}_\delta  \lambda$ such that:
\begin{enumerate}
    \item $\mathcal{U}\cap N\in N$
    \item $\mathcal{P}_\delta \lambda \cap N \in \mathcal{U}$.
\end{enumerate}

\begin{conjecture}
[Woodin's Ultimate-$L$ Conjecture]\label{WoodinUlt}
Suppose that $\delta$ is an extendible cardinal. Then there is an inner model $N\subseteq HOD$ such that:
\begin{enumerate}
    \item $N$ is a weak extender model for the supercompactness of $\delta$.
    \item $N\models$`$V$=Ultimate-$L$'.
\end{enumerate}
\end{conjecture}

\noindent
Now, crucially for our purposes, if the Ultimate-$L$ Conjecture holds, then letting $T$ be the theory $\mathsf{ZFC}+$`There exists a proper class of extendible cardinals' we have that $\mathsf{MV}_T$ proves that the core has an inner model $\mathcal{V}$ which is contained in HOD, and satisfies  $V$=Ultimate-$L$. Moreover, by Woodin's Universality Theorem,\footnote{\cite{woodin2017}, p. 16, Theorem 3.26.} $\mathcal{V}$ satisfies that there exists a proper class of extendible cardinals. However, $\mathcal{V}$ need not be a world itself, but, if it is so, then $\mathcal{V}$ is the \textit{core}. So, here follows the first  result of this subsection:

\begin{proposition}
 Let $T$ be $\mathsf{ZFC}+$`There exists a proper class of extendible cardinals'. Then $\mathsf{MV}_T$ proves that the following are equivalent:
\begin{enumerate}
    \item There exists a world satisfying `$V$=Ultimate-$L$'
    \item $\mathcal{C}\models$`$V$=Ultimate-$L$'.
\end{enumerate}
\end{proposition}

 \begin{proof}
Let $M^G$ be a model of $\mathsf{MV}_T$, and let $\mathcal{W}$ be a world in $M^G$ that satisfies $V$=Ultimate-$L$. Then $\mathcal{W}$ is a ground. But $\mathcal{W}$ has no proper grounds, hence since $MV_T$ implies that the core exists and is the mantle, $\mathcal{W}$ must be contained in the core, and therefore it must be the core.
\end{proof}

The following two propositions nail down the consistency of $V$=Ultimate-$L$ with $\mathsf{MV}_T$ (where $T=\mathsf{ZFC}$+`there exists a class of extendible cardinals'), assuming $V$=Ultimate-$L$'s own consistency.

\begin{proposition}\label{Ult-L}
 Let $T$ be $\mathsf{ZFC}+$`There exists a proper class of extendible cardinals'. If $T+$`$V=\mbox{Ultimate}$-$L$'  is consistent, then so is $\mathsf{MV}_T$ plus `$\mathcal{C}\models$ T+$V$=Ultimate-$L$'.
\end{proposition}

\begin{proof}
Let $M$ be a model of $T+$`$V=\mbox{Ultimate}$-$L$'. Then for every $G\subseteq Coll(\omega , <Ord)^M$-generic filter over $M$, $M^G$ satisfies `$\mathcal{C}\models V=$Ultimate-$L$', by last Proposition.
\end{proof}

\begin{proposition}
Assume the Ultimate-$L$ Conjecture. If $T=\mathsf{ZFC}+$`There exists a proper class of extendible cardinals' is consistent, then so is $\mathsf{MV}_T+$`$\mathcal{C}\models V$=Ultimate-$L$'.
\end{proposition}

\begin{proof}
Let $M$ be a model of $T$. Let $\delta$ be the least extendible cardinal in $M$.  By the Ultimate-$L$ Conjecture, $M$ has an inner model $N$ satisfying $V=$Ultimate-$L$ and which is a weak extender model for the supercompactness of $\delta$. By the Universality Theorem, in $N$ there is a proper class of extendible cardinals. Now let $G\subseteq Coll(\omega , <Ord)^N$ be generic over $N$. Then $N^G$ is a model of $\mathsf{MV}_T$ plus $\mathcal{C}\models$ $``V=$Ultimate-$L"$.
\end{proof}

However, also the negation of $V$=Ultimate-$L$, as shown by the following proposition, is consistent with $\mathsf{MV}_T$:

\begin{proposition}\label{nUlt-L}
If the theory $T=\mathsf{ZFC}+$`There exists a proper class of extendible cardinals'+`$V$=Ultimate-$L$' is consistent, then so is $\mathsf{ZFC}+$`There exists a proper class of extendible cardinals'+`$\, {\mathcal C} \not \models V$=Ultimate-$L$'.
\end{proposition}

\begin{proof}
Let $M$ be a model of $T$. Force GA by class forcing, as in \cite{reitz2007}, so that (by \cite{bagaria-poveda2020}) in the resulting model $N$ there is a proper class of extendible cardinals. If $G\subseteq Coll(\omega , <Ord)^N$ is generic over $N$, then $N^G$ is a model of $\mathsf{MV}_T$ plus $\mathcal{C}=N\not \models`V$=Ultimate-$L$'.
\end{proof}

Similar considerations apply to the case where the core is a weak extender model for $\delta$ supercompact. If the theory $T=\mathsf{ZFC}+$`There exists a proper class of extendible cardinals'$+$ `There exists a weak extender model for $\delta$ supercompact' is consistent, then so is $T$ plus the core is such a model, and also $T$ plus the core isn't.

\subsection{The Role of Ultimate-$L$ within $\mathsf{MV}$}

From what Steel says in \cite{steel2014}, it seems reasonable to assume that he would encourage the adoption of $\mathsf{ZFC}$+LCs+$V$-Ultimate-$L$ as the Core Universist's ultimate theory. It is crucial to make clear how this could possibly happen.

Steel declares $V$=Ultimate-$L$ to be the right axiom for the core for the following reasons:\footnote{\cite{steel2014}, p. 169.} (1) it implies $V=\mathcal{C}$; it implies the existence of a `fine structure theory' for the core; (2) it is consistent with all LCs.\footnote{\label{AxiomH}At the time of the writing of \cite{steel2014}, $V$=Ultimate-$L$ had yet to be formulated as a mathematical axiom. The axiom that Steel advocated in that work was, in fact, Woodin's Axiom $\mathsf{H}$, in turn, a version of $V$=HOD. Now, assuming $V$=Ultimate-$L$, then the Axiom $\mathsf{H}$ also holds. However, the reasons Steel adduces as evidence in favour of the Axiom $\mathsf{H}$ are exactly the same as he might adduce in favour of $V$=Ultimate-$L$.} But, as we know, in view of $\mathsf{MV}$'s own machinery and goals, these motivations may just be seen as auxiliary arguments for the acceptance of $V$=Ultimate-$L$, as our prospects to settle on $V$=Ultimate-$L$ as the right extension of $\mathsf{ZFC}$ entirely depend, by $\mathsf{MV}$'s own lights, on detecting its presence in $\mathcal{L}_{\mathsf{MV}}$.

So we proceed to assess these prospects by resuming the narrative about the Core Universist's possible choices laid out at the beginning of section 5.\footnote{For salient points of the account which follows, we are also indebted to \cite{maddy-meadows2020}, section 6.}

As said there, the Universist might tentatively interpret the $\mathcal{L}_{\in}$-translate of the $\mathcal{L}_{\mathsf{MV}}$-sentence `the core exists' as suggesting (`practically implying') the view that $V$ \textit{is} the core. Syntactically, this would correspond to adopting the theory $\mathsf{ZFC}+$LCs+$V=\mathcal{C}$. So, our Universist could, provisionally, settle on such a theory as the ultimate extension of $\mathsf{ZFC}$. However, as we know, the core is a highly \textit{indeterminate} object, so $V=\mathcal{C}$ wouldn't be very informative. By contrast, if our Universist interpreted the $\mathcal{L}_{\in}$-translate of `the core exists' as suggesting that $V$ is Ultimate-$L$, then she would be able to \textit{determinately} fix the features of the core, and would get $V=\mathcal{C}$ as a bonus.

To be more precise, in the presence of such an interpretation, (Transl) would guarantee that:

\[CH \leftrightarrow t(CH^{Ultimate-L}) \]

\noindent
and, more generally, that, for all $\varphi$, if $\varphi$ isn't settled by $T= \mathsf{ZFC}+$LCs:

\[ \varphi \leftrightarrow t(\varphi^{Ultimate-L}) \]

\noindent
So far so good, but is this strategy really workable? In fact, as we would like to point out, there may be severe hindrances in the way of its execution. First of all, if one looks at the functioning of (Transl), the $\mathcal{L}_{\mathsf{MV}}$-sentence `the core exists', doesn't literally translate to the $\mathcal{L}_{\in}$-sentence `$V$ is the core', let alone to `$V$=Ultimate-$L$' (see Definition \ref{Transl}). Therefore, some less quick, and more subtle kind of reasoning might underlie the adoption of $V=\mathcal{C}$ on the Universist's part, and this may have to do with the definability of the \textit{core} in $\mathsf{MV}$. As we know, if $\mathsf{MV}$ has a definable world, then it has a unique definable world (cf. Theorem \ref{defworld}). Now, we know that the core is definable, and since there is just \textit{one} such object in $\mathsf{MV}$, $\mathsf{MV}$'s unique definable world is the core itself. Therefore, the Core Universist might thrive on considerations about the \textit{definability} of the core within $\mathsf{MV}$, and see these as strongly suggesting that $V=\mathcal{C}$. But even if this argument ultimately persuaded us to adopt $V=\mathcal{C}$, it wouldn't be sufficient to persuade us to adopt $V$=Ultimate-$L$, anyway.

What we'd need here to make the $\mathcal{L}_{\mathsf{\in}}$-translate of the $\mathcal{L}_{\mathsf{MV}}$-sentence `the core exists' suggest (imply) `$V$=Ultimate-$L$', would be some strengthening of $T$ in the theory $\mathsf{MV}_T$. As we shall see, unfortunately, all such strengthenings will produce results which are not fully satisfactory.

Two main cases are possible. First, take $T$ to be theory $\mathsf{ZFC}$+LCs+$V$= Ultima-te-$L$, and consider $\mathsf{MV}_T$. The theory could, potentially, do the required job, but it is inconsistent! The reason is, any model of such theory would violate Axiom 4 of $\mathsf{MV}$, as no forcing extension of $\mathsf{MV}$ could be a world satisfying $T$.
A more modest strengthening of $T$, on the other hand, might suit our purposes. Take $T$ to be: $\mathsf{ZFC}$+LCs+`$\mathcal{C}\models$V=Ultimate-$L$', then the $\mathcal{L}_{\mathsf{\in}}$-translate of `the core is Ultimate-$L$' would be a lot closer to what we'd need to have at hand, but, still, the translation wouldn't automatically imply `$V$ is the core'. But there's also another trouble with this choice of $T$: the axiom `$\mathcal{C}\models$V=Ultimate-$L$' wouldn't, \textit{prima facie}, seem to be justified on the grounds of $\mathsf{MV}$'s very evidential framework, so the only reason why one would add it to $T$ in $\mathsf{MV}_T$ would have to do with other, `external', reasons, such as, (1.)-(3.) above, or the prior belief in the correctness of $V$=Ultimate-$L$, but now relying on such reasons, somehow, would beg the question of why the Core Universist should adopt $V$=Ultimate-$L$.

But maybe there are ways to make `$\mathcal{C}\models$V=Ultimate-$L$' justified on $\mathsf{MV}$'s very evidential framework. We proceed to examine arguments potentially to that effect in the next subsections.

\subsection{Woodin's Argument for Ultimate-$L$}

The first argument derives from Woodin's own reflections on the philosophical aspects of Conjecture \ref{WoodinUlt} (Ultimate-$L$ exists).

In \cite{woodin2011b}, Woodin has proposed to construe the Inner Model Programme as the expression of the fundamentally \textit{non-formalistic} character of set theory. Woodin's Argument, as we shall call it, first introduces two different positions about set-theoretic truth, the Skeptic's and the Set-Theorist's.

\begin{itemize}
    \item \textit{Skeptic}: set-theoretic theorems are truths about finitary objects (\textit{proofs}).

    \item \textit{Set-theorist}: set-theoretic theorems are truths about an \textit{existing} realm of mathematical objects.
\end{itemize}

Now, Woodin argues that the dialectic between the Skeptic and the Set-Theorist reaches its climax on the issue of the \textit{consistency} of LCs: the Skeptic holds that the consistency of LCs is a purely \textit{finitistic} fact, whereas the Set-Theorist believes that it depends on `thick' intuitions about universes which contain them, in particular, \textit{canonical inner models}. In order to further articulate the Set-Theorist's position, Woodin formulates the following principle:

\vspace{11pt}

\noindent
\textbf{Set-Theorist's Cosmological Principle (SCP)}. A Large Cardinal Axiom is \textit{consistent} if and only if there is an inner model which satisfies it; the prediction of its consistency is true, because LCAs are \textit{true}.

\vspace{11pt}

\noindent
Woodin also indicates concrete ways in which SCP might be disconfirmed, that is, by (i) finding a \textit{proof} of the consistency/inconsistency of one specific LCA, or, more subtly (a strategy that Woodin calls Skeptic's Retreat) by showing that: (ii) the consistency (or inconsistency) of one LCA \textit{formally entails} the consistency (inconsistency) of \textit{all} LCAs.\footnote{Cf. \cite{woodin2011b}, pp. 452ff. As regards how the SCP might be disconfirmed, Woodin's text is rather terse; however, it seems plausible to conjecture that Woodin's misgivings are directed at the fact that a proof of the consistency/inconsistency of \textit{one} LCA using inner models, might ultimately generalise to a proof of the consistency of \textit{other}, even \textit{all}, LCAs, something which would trivialise the Set-Theorist's view about the structure of consistency predictions.} Woodin then proceeds to make considerations about the possibility that, if the Ultimate-$L$ Conjecture is true, then we may have a scenario where neither (i) nor (ii) would be applicable.

We shall entirely leave aside the issue of the plausibility of Woodin's Argument, and exclusively focus on the potential usefulness of the argument for our Core Universist. The argument revolves around the idea that our belief in the correctness of \textit{all} LCs is based on the belief that they are consistent, in turn, via SCP, that they have \textit{inner models}. Thus, Woodin's Argument practically suggests that, if one sees LCs as the `right' extensions of $\mathsf{ZFC}$, it is because one views them as consistent via \textit{inner models}. So, the reasoning goes, once one commits oneself to \textit{all} LCs, as the $\mathsf{MV}$ supporter does, then, there is a (thick) sense in which one also commits oneself to Ultimate-$L$, that is, to Conjecture \ref{WoodinUlt}. But then, if Ultimate-$L$ exists, it should be a \textit{definable} world and, thus, based on Theorem \ref{defworld}, it would be the \textit{core}; the correctness of `$\mathcal{C} \models$V=Ultimate-$L$' would thus finally be vindicated, seemingly, on $\mathsf{MV}$'s own evidential grounds.

However, the argument has two main problems for our $\mathsf{MV}$-based Core Universist.

The first one is that the evidential resources it invokes (truth, consistency predictions, etc.) are not, in fact, available to her. Steel's conception prides itself on not being dependent on any `metaphysics' of universes. Therefore, intuitions about the structure of models with LCs, which underlie Woodin's Argument and SCP, if argumentatively efficacious, may not bear on our Core Universist's acceptance of `$\mathcal{C}\models$Ultimate-$L$'.

The other one is that, by using Woodin's Argument in the present context, then the Core Universist would become practically indistinguishable from the Classic Universist who supports $V$=Ultimate-$L$: even coming from slightly different backgrounds, both would, indeed, agree on the fact that there is one \textit{universe} of set theory which is preferable to all others (Ultimate-$L$) from the beginning, and it does not seem that the $\mathsf{MV}$ supporter's evidential framework would commit her to such a view.

Let us now move on to explore another potential justificatory strategy.

\subsection{An Extrinsic Argument}

A different argument is based on resuming Steel's reasons to adopt $V$=Ultimate-$L$ (1.)-(3.) stated in section 5.2: these, overall, suggest to the Core Universist that $V$=Ultimate-$L$ is a very successful axiom, which could be seen as being \textit{already} justified `extrinsically' -- insofar as practically \textit{all} undecidable statements are settled by it.

More precisely, the Core Universist might use a form of `regressive' reasoning here: she will require $\mathsf{MV}_T$ to incorporate `$\mathcal{C} \models$V=Ultimate-$L$' in $T$, because she takes $V$=Ultimate-$L$' to be an \textit{already} independently justified axiom.

However, in section 5.2, we have already hinted at the inherent limitations of this strategy: $\mathsf{MV}$ would no longer be used as a means to indicate which bits of $\mathcal{L}_{\in}$ are meaningful; on the contrary, it would be $\mathcal{L}_{\in}$ to provide us information about what specific $T$ should be chosen in $\mathsf{MV}_T$. In other terms, the Core Universist wanting to use this `extrinsic' argument would, ultimately, violate her own unbiasedness about all different theories expressed by $\mathsf{MV}$.

Another, more general, concern is that $V$=Ultimate-$L$ isn't the only successful axiom she has at hand. In particular, since she knows that the core need not be Ultimate-$L$, and that $T$ in $\mathsf{MV}_T$ is consistent with  `$\mathcal{C} \not \models$V=Ultimate-$L$' (Theorem \ref{nUlt-L}), the Core Universist could ultimately settle on other, equally successful, axioms for the core. For instance, she might want to adopt `$\mathcal{C} \models$ MM$^{++}$', as MM$^{++}$ implies that $c=\aleph_2$, and is clearly able to settle many other undecidable statements.

One final worry about the argument, which is worth mentioning, is that `success' wasn't really part of Steel's narrative concerning $\mathsf{MV}$ from the beginning, although, clearly, LCs may be interpreted as being very `successful' axioms. We do not want to delve into the full intricacies of the topic, but `success' may really be a very volatile criterion, which, although helpful, may not lead the Core Universist to make ultimate choices about the nature of $\mathsf{MV}$.

\subsection{Summary}

Let's take stock. The progression from $\mathsf{ZFC}$ to $T=\mathsf{ZFC}$+LCs+`$\mathcal{C} \models$V=Ultimate-$L$', summarised in the table below, shows that ascending through \textit{interpretative power} (and \textit{consistency strength}) of theories, is, presently, insufficient to suggest to the Core Universist that $V$ is the core, or that $V$ is any specific `world', for instance, Ultimate-$L$. As we have seen, on the one hand, by adding further hypotheses to $T$ in $\mathsf{MV}_T$, one may get an inconsistent theory; on the other, one could make choices, such as the addition of `$\mathcal{C} \models$V=Ultimate-$L$', which, however, on $\mathsf{MV}$'s own evidential grounds, do not seem to be much justified. This, overall, leaves us with the following, somewhat unpalatable, dilemma: either to stay with $T=\mathsf{ZFC}$+LCs, in $\mathsf{MV}_T$ and, thus, view the core as fundamentally indeterminate, or move to a, globally, less justified strengthening of $T$, but finally get a \textit{determinate} core.

\medskip

\begin{table}[h]

\footnotesize

\begin{tabular}{||c c c c||}
 \hline
 Theory & Core (Existence) & Core (Determinacy) & Suggests $V=\mathcal{C}$ \\ [0.5ex]
 \hline
 ZFC & No & / & /\\
 \hline
 ZFC+LCs & Yes & No & Yes?\\
 \hline
 ZFC+LCs+$V$=Ult-$L$ & \textbf{Inconsistent!} &  & \\
 \hline
 ZFC+LCs+`$\mathcal{C} \models$V=Ultimate-$L$' & Yes & Yes & Yes?\\
 \hline
 ... & ... & ... &...\\ [1ex]
 \hline

\end{tabular}

\caption{Behaviour of different theories $T$ with respect to the core.}

\end{table}

\medskip

\normalsize

\section{Concluding Remarks and Ways Forward}

Based on what one can prove in the theory $\mathsf{MV}_T$, where $T=\mathsf{ZFC}$+`there exists a proper class of extendible cardinals', and some potential strengthenings, we have assessed the prospects of what we have called Steel's Programme, and of the corresponding philosophical position that Steel has called Weak Absolutism, and we have called Core Universism.

Our tentative conclusion is that there might still be a long way to go before the view that the core is Ultimate-$L$ and, consequently, that $V$=Ultimate-$L$ is the ultimate axiom for the Core Universist, gets fully validated; meanwhile, as is clear, $V$=Ultimate-$L$ might ultimately be accepted on entirely different grounds.

We conclude the paper with suggesting some possible future scenarios for the investigation of the `core hypothesis' within $\mathsf{MV}$:

\vspace{11pt}

\noindent
\textbf{First Scenario}. Stronger, hitherto unknown, LCs will, `more determinately', settle the features of the core.

\vspace{11pt}

\noindent
The idea, here, is that further ascending through consistency strength in an ideally `richer' large-cardinal hierarchy will help settle the features of the core. Clearly, at present, nobody could predict whether there will be concrete developments in this direction, but note that Usuba's key theorem (Theorem \ref{usuba}) has set an interesting precedent for results in this area which were widely unexpected.

\vspace{11pt}

\noindent
\textbf{Second Scenario}. The `core hypothesis' will ultimately be taken to just be a theoretical tool to foster and study the interplay between \textit{multiverse} and \textit{universe thinking} while dealing with set-theoretic incompleteness.

\vspace{11pt}

\noindent
The scenario above construes Steel's efforts as going in the direction of clarifying, not prescribing solutions for, the issues of whether there is a core universe and of what the core should be like. In this scenario, the core hypothesis would not, \textit{per se}, suggest a unique course of action for Universists, yet may be taken to be broadly informative on what resources and additional hypotheses are needed if one wants to settle the undecidable statements.\footnote{This interpretation is, arguably, already inherent in Steel's declared goal of using $\mathcal{L}_{\mathsf{MV}}$ `to trim the syntax of $\mathcal{L}_{\in}$' and `thus avoiding asking pseudo-questions'. Cf. \cite{steel2014}, p. 168.}

\vspace{11pt}

\noindent
\textbf{Third Scenario}. The issue of the existence of a `preferred universe' will be declared to be \textit{insoluble} on purely proof-theoretic grounds, that is, by just detecting traces of \textit{multiverse language} in the \textit{language of set theory}, whilst other conceptual \textit{resources} will ultimately be seen as fundamental to solve it.

\vspace{11pt}

\noindent
As we have seen, Woodin's Argument, for instance, provides us with alternative resources to solve the issue of what counts as a `preferred' universe. Now, one could conjecture that it will turn out to be inevitable to resort to this kind of arguments to meet the Core Universist's requirements.

In the paper, we have also addressed and, by our lights, made more transparent, several features, both mathematical and philosophical, of $\mathsf{MV}$: in particular, the proof of the existence of the core, its persistent indeterminacy over different choices of $T$ for $\mathsf{MV}_T$, the mathematics of Ultimate-$L$ and $\mathsf{MV}$, the justifiability and role of LCs. Now, further mathematical work is, already at this stage, needed in order to further assess $\mathsf{MV}$'s hypotheses. Any (incomplete) description of such work should, at least, include the following items:

\begin{itemize}
    \item The status of the Ultimate-$L$ Conjecture

    \item The prospects of a unified account of LCs

    \item A better understanding of the notion of `canonicity' with respect to (alternative) models of set theory
\end{itemize}

We expect that further enlightenment on these issues will also carry with itself a more detailed understanding of what the Core Universist may legitimately claim, based on the structure of the multiverse. But, as we have seen, definite answers to many issues concerning $\mathsf{MV}$, some confirming, others disconfirming, the Core Universist's expectations, can already be provided.

\pagebreak

redpajama_set_name: RedPajamaArXiv


dhw: If God exists, I would also assume he is capable of doing whatever he wants. I just don't know on what authority you can tell us that he didn't want to give cells autonomous intelligence.
Because I think cells are automatic.
dhw: "Agree"? Her statement could hardly be clearer: the intelligence is their own: they register and process information and use their conscious intelligence to take their decisions. She is NOT saying that God plants the info and plans their responses, which you claim is the only way you can see for cells to act intelligently. So who are you agreeing with?


Amherst MA — Just minutes after his thesis deadline had passed, Psychology major Ryan Strobl '15 regretfully informed his advisor, Professor Catherine A. Sanderson, as well as the rest of the Amherst Psychology Department, that he was a "bit too preoccupied" working on this thesis to turn in his thesis right now.
"It's coming along really well, though," he added.
This entry was posted in NewsBrief, Thoughts on Theses by Chief Amherst Correspondent. Bookmark the permalink.


REDCOM attends numerous industry trade shows, conferences, and events around the world. The list below outlines recent and upcoming events. Hopefully, we'll see you there!
The REDCOM team will be in Chicago during the 2019 Technology & Business Solutions eXchange Conference for private meetings, discussions, and networking. Be sure to sign up now to secure your appointment with REDCOM at the show!
The REDCOM team will be in Columbus during the 2019 OTA Convention for private meetings, discussions, and networking. Be sure to sign up now to secure your appointment with REDCOM at the show!
The REDCOM team will be in Uncasville during the 2019 Southern New England Design-to-Part Show for private meetings, discussions, and networking. Be sure to sign up now to secure your appointment with REDCOM at the show!
The Special Operations Forces Industry Conference (SOFIC) is the premier conference for the SOF community to interact with industry and to collaborate on the challenges, initiatives, and way-ahead in delivering the most cutting-edge capabilities into the hands of SOF operators. REDCOM will be partnering with Mercury Systems at this show. Join us in their booth #1102.
Canto is the leading authority in shaping information, communication, and technology in the Caribbean region. Be sure to visit with REDCOM's Jose Pardo to learn about REDCOM's communications solutions for the Service Layer.


text:
Yemen conflict: UN experts detail possible war crimes by all parties
UN human rights experts believe war crimes may have been committed by all parties to the conflict in Yemen.
In their first such report, they allege Yemeni government forces, the Saudi-led coalition backing them, and the rebel Houthi movement have made little effort to minimise civilian casualties.
They point to attacks on residential areas in which thousands have died.
The warring parties are also accused of arbitrary detentions, torture, enforced disappearances and recruiting children.
The experts will present their report to the UN Human Rights Council next month.
Why is there a war in Yemen?
Yemen has been devastated by a conflict that escalated in early 2015, when the Houthis seized control of much of the west of the country and forced President Abdrabbuh Mansour Hadi to flee abroad.
Alarmed by the rise of a group they saw as an Iranian proxy, the UAE, Saudi Arabia and seven other Arab states intervened in an attempt to restore the government. They have received logistical and intelligence support from the US, UK and France.
At least 6,660 civilians have been killed and 10,563 injured in the fighting, according to the United Nations. Thousands more civilians have died from preventable causes, including malnutrition, disease and poor health.
The fighting and a partial blockade by the coalition has also left 22 million people in need of humanitarian aid, created the world's largest food security emergency, and led to a cholera outbreak that is thought to have affected 1.1 million people.
The so-called Group of Experts note that coalition air strikes have caused most direct civilian casualties, and that they have hit residential areas, markets, funerals, weddings, detention facilities, civilian boats and medical facilities.
The report says they "have reasonable grounds to believe that individuals in the government of Yemen and the coalition may have conducted attacks in violation of the principles of distinction, proportionality and precaution that may amount to war crimes"
They add that the naval and air restrictions imposed by the coalition on rebel-held areas to halt alleged weapons smuggling may also constitute a violation of the proportionality rule of international humanitarian law, while the effective closure of Sanaa airport may violate the principle of protection for the sick and wounded.
The experts also expressed concern at the situation in the southern city of Taiz, where the Houthis have been besieging a government-held area for three years.
The report says civilians, including women and children, have been hit by shelling and snipers from the Houthis and other parties while in their homes, fetching water at local wells, or on their way to purchase food or seek medical attention.
The Houthis are accused of indiscriminately using "weapons with wide area effect" in Taiz and other urban settings, which would constitute a war crime. The experts also found evidence of widespread arbitrary detention throughout the country by all parties, and ill-treatment and torture of some facilities.
Victims and witnesses also described to the experts "persistent and pervasive aggressive behaviour", including sexual violence perpetrated by a pro-government force known as the Security Belt and UAE personnel, according to the report.
The experts say they also received information indicating all parties had conscripted or enlisted children – some as young as eight years old – and used them to participate actively in hostilities. There was no immediate response from the coalition, Yemen's government, or the Houthis. But the BBC's Imogen Foulkes in Geneva says Saudi Arabia did not want this inquiry, and will not like its findings. On Monday, the coalition accused the UN team in Yemen of bias after it called for expressing condolences to the families of more than 30 civilians, including 22 children, reportedly killed in two air strikes last week.
The experts say they have identified, where possible, individuals who may be responsible for war crimes and passed a confidential list of their names to the UN high commissioner for human rights.
They also urged the international community to "refrain from providing arms that could be used in the conflict" – an apparent reference to Western countries, who are selling weapons to the coalition, and to Iran, which the coalition alleges is smuggling weapons the Houthis.
"Despite the severity of the situation, we continue to witness the total disregard for the suffering of the people of Yemen. This crisis has reached its peak with no apparent sight of light at the end of the tunnel," Charles Garraway, one of the experts, told a news conference.
Soy boom devours Brazil's tropical savanna
Nestle, Starbucks wrap up $7.15 billion licensing deal

redpajama_set_name: RedPajamaCommonCrawl


If you are an OpenStack contributor, you likely rely on DevStack for most of your work. DevStack is, and has been for a long time, the de-facto platform that contributors use for development, testing, and reviews. In this article, I want to introduce you to a project I'm a contributor to, called openstack-ansible. For the last few months, I have been using this project as an alternative to DevStack for OpenStack upstream development, and the experience has been very positive.
Federation support in OpenStack has been greatly improved in the Kilo release, and, while there are still some rough edges, the feature is now usable, if you don't mind adding a little elbow grease. This article focuses on one specific configuration called Keystone-to-Keystone federation (or K2K), in which users of an OpenStack cloud use their credentials to access services in another OpenStack cloud, with both clouds using Keystone as their identity services.
This is the fourth and last article in my series on OpenStack orchestration with Heat. In the previous articles, I gave you a gentle introduction to Heat, and then I showed you some techniques to orchestrate the deployment of single and multiple instance applications on the cloud, all done with generic and reusable components.
Today I'm going to discuss how Heat can help with one of the most important topics in cloud computing: scalability. Like in my previous articles, I'm going to give you actual examples that you can play with on your OpenStack cloud, so make sure you have an environment where you can run tests, whether it's a Rackspace Private Cloud, DevStack or any other OpenStack distribution that includes Heat.
This is the third article in my series on OpenStack orchestration with Heat. In Part 1, I introduced the HOT template syntax, and then in Part 2, I showed you some of the techniques Heat offers to orchestrate the deployment of applications that run entirely within a single compute instance.
Today, building on the same ideas exposed in my previous article, I'm going to show you how to design deployments across more than one instance, and I'm going to demonstrate these concepts by deploying an application that runs on a server and connects to a MySQL database on another server. You have seen how to deploy a Python application in my previous examples, so, to add some variety, I'm now going to switch to a PHP application as guinea pig. That application is none other than the venerable Wordpress.
Welcome to the second part of my series on OpenStack orchestration with Heat. In the previous article, I gave you an introduction to Heat orchestration. All the examples that I showed you were simple and not terribly useful, as they were only intended to introduce the structure of the HOT (Heat Orchestration Template) syntax.
In today's article, I'm going to elevate the complexity quite a bit, demonstrating some of the tricks you can use with Heat to perform deployments of single instance applications. As with the introductory examples, you are encouraged to try my examples on a Rackspace Private Cloud, DevStack or any other OpenStack installation that includes Heat.
With this article I begin a series of hands-on developer oriented blog posts that explore OpenStack orchestration using Heat.
To make the most of this article, I recommend that you have an OpenStack installation where you can run the examples I present below. You can use our Rackspace Private Cloud distribution, DevStack, or any other OpenStack distribution that includes Heat.