5.1 Partial Morleyizations

Notation 5.1 Given a signature $\tau $ , let $\phi (x_0,\dots ,x_n)$ be a $\tau $ -formula.

We let:

• • $R_\phi $ be a new $n+1$ -ary relation symbols,

• • $f_\phi $ be a new n-ary function symbols,Footnote ¹⁹

• • $c_\tau $ be a new constant symbol.

We also let

$$\begin{align*}{\mathsf{AX}}^0_\phi:= \forall\vec{x}[\phi(\vec{x})\leftrightarrow R_\phi(\vec{x})], \end{align*}$$

$$ \begin{align*} {\mathsf{AX}}^1_\phi:= &\forall x_1,\dots,x_n\,\\ &[(\exists! y\phi(y,x_1,\dots,x_n)\rightarrow \phi(f_\phi(x_1,\dots,x_n),x_1,\dots,x_n))\wedge\\ &\wedge(\neg\exists! y\phi(y,x_1,\dots,x_n)\rightarrow f_\phi(x_1,\dots,x_n)=c_\tau)] \end{align*} $$

for $\phi (x_0,\dots ,x_n)$ having at least two free variables, and

$$\begin{align*}{\mathsf{AX}}^1_\phi:= \left[ (\exists! y\phi(y))\rightarrow \phi(f_\phi) \right]\wedge \left[ (\neg\exists! y\phi(y))\rightarrow c_\tau=f_\phi \right] \end{align*}$$

for $\phi (x)$ having exactly one free variable.

Let $\mathsf {Form}_\tau $ denotes the set of $\tau $ -formulae. For $A\subseteq \mathsf {Form}_\tau \times 2$ :

• • $\tau _A$ is the signature obtained by adding to $\tau $ all relation symbols $R_\phi $ , for $(\phi ,0)\in A$ , and all function symbols $f_\phi $ , for $(\phi ,1)\in A$ (together with the special symbol $c_\tau $ if at least one $(\phi ,1)$ is in A).

• • $T_{\tau ,A}$ is the $\tau _A$ -theory having as axioms the sentences ${\mathsf {AX}}^i_\phi $ for $(\phi ,i)\in A$ .

We let $\tau ^*=\tau _C$ for $C=\mathsf {Form}_\tau \times \left \lbrace 0 \right \rbrace $ and $T^*$ be the $\tau ^*$ -theory $T_{\tau ,C}$ .

Roughly speaking the axioms ${\mathsf {AX}}^i_\phi $ are meant to formalize the idea that the (potentially complicated) concept formalized by $\phi $ according to $\tau $ becomes an elementary concept if we formalize instead it using $\tau _{\left \lbrace (\phi ,i) \right \rbrace }$ . More precisely, given a $\tau $ -theory S:

• • ${\mathsf {AX}}^0_\phi $ transforms the concept of S formalized by the $\tau $ -formula $\phi $ (which could have a high Levy complexity according to $\tau $ ) in a concept formalized by an atomic $\tau \cup \left \lbrace R_\phi \right \rbrace $ -formula according to $S+{\mathsf {AX}}^0_\phi $ .

• • For any $\tau $ -theory S, and for $C=\mathsf {Form}_\tau \times \left \lbrace 0 \right \rbrace $ , $S^*=S+T_{\tau ,C}$ is a $\tau ^*=\tau _C$ -theory admitting quantifier elimination (the Morleyization of T).Footnote ²⁰

• • ${\mathsf {AX}}^1_\phi $ performs a delicate Skolemization process. The standard Skolemization introduces for each formula $\phi $ a Skolem term $f_\phi $ together with an axiom stating $\forall \vec {x}\,[\exists y\,\phi (\vec {x},y)\leftrightarrow \phi (\vec {x},f_\phi (\vec {x}))]$ . However, this has serious drawbacks. Consider the following situation: we have a $\tau $ -structure $\mathcal {M}$ which is such that for all $b\in \mathcal {M} \exists y\,\phi (b,y)$ holds, and also such that for some $a\in \mathcal {M}$ , we have that $\phi (a,y)$ has more than one solution in $\mathcal {M}$ . In this case we can find distinct functions $h_0,h_1:\mathcal {M}\to \mathcal {M}$ such that $\forall x\,[\exists y\,\phi (x,y)\leftrightarrow \phi (x,f_\phi (x))]$ is satisfied in the expansions of $\mathcal {M}$ to a $\tau \cup \left \lbrace f_\phi \right \rbrace $ -structure which interprets $f_\phi $ by either one of the $h_i$ . Which one of the two expansions should we consider the natural Skolemization of $\mathcal {M}$ for the formula $\phi $ ? ${\mathsf {AX}}^1_\phi $ rules out this source of ambiguity by saying that when $\phi (a,y)$ has more than one solution in $\mathcal {M}$ , then $f_\phi ^{\mathcal {M}}$ does not select any of them and instead behaves trivially by assigning to a the value $c_\tau ^{\mathcal {M}}$ . In this way, given a formula $\psi (\vec {x},y)$ , once we choose the interpretation of $c_\tau $ , any $\tau $ -structure has a unique expansion to a $\tau \cup \left \lbrace f_\psi ,c_\tau \right \rbrace $ -structure which is a model of ${\mathsf {AX}}^1_\psi $ and has that interpretation of $c_\tau $ .

Notice that with respect to the signatures expanding $\left \lbrace \cdot ,+,0,1 \right \rbrace $ discussed in the previous section, axiom (5) is (equivalent to) ${\mathsf {AX}}^1_\phi $ for $\phi (x,y)$ being $x\cdot y=1$ (assuming $c_\tau $ is interpreted by $0$ ). Similarly axiom (6) is (equivalent to) ${\mathsf {AX}}^1_\psi $ for $\psi (x,y)$ being $x+ y=0$ . In what follows we want to analyze what happens when the Morleyization process is performed on arbitrary subsets of $\mathsf {Form}_\tau \times 2$ .

5.2 Absolute model companionship

The following properties will bring us to introduce the notion of absolute model companionship.

• • A $\tau $ -structure $\mathcal {M}$ is T-ec if and only if it is $T^\tau _\forall $ -ec (by [Reference Viale43, fact 8]).

• • If T is complete, a T-ec structure $\mathcal {M}$ realizes any $\Pi _2$ -sentence which is consistent with $T^\tau _\forall $ (by [Reference Viale43, fact 12]).

• • If a $\tau $ -theory S is the model companion of a $\tau $ -theory T, S is axiomatized by its $\Pi _2$ -consequences for signature $\tau $ (by Robinson’s test, e.g., [Reference Viale43, lemma 14]).

These results are outlining “syntactic properties” of T-ec models which reflect the “semantic fact” that T-ec models are the “closed-off” structures for T; furthermore they outline that the axiomatization of the model companion of a complete theory can be detected by separately solving a consistency task for each $\Pi _2$ -sentence for $\tau $ . One might therefore wonder whether these syntactic properties can give another equivalent characterization of model companionship, i.e., if the model companion S of a $\tau $ -theory T (whenever it exists) can be axiomatized by the family of all $\Pi _2$ -sentences for $\tau $ which holds in some model of the universal and existential fragment of T. This is not always the case; it holds when T is a complete model companionable theory, however there is a standard counterexample: the $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -theory $\mathsf {Fields}_0$ of fields of characteristic $0$ , given by axioms (1)–(3) and the infinitely many axioms granting that the characteristic of its models is $0$ (note that $\mathsf {Fields}_0$ is not a complete theory) has as its model companion $\mathsf {ACF}_0$ , which adds axiom (4) to $\mathsf {Fields}_0$ (note that $\mathsf {ACF}_0$ is a complete theory). Indeed, the sentence $\forall x \neg (x^2+1=0)$ is a $\Pi _2$ -sentence (actually $\Pi _1$ ) which is not in $\mathsf {ACF}_0$ but holds in $\mathbb {Q}$ , hence is consistent with the universal and existential fragment of $\mathsf {Fields}_0$ .

Absolute model companionship rules out counterexamples of this kind to the above tentative characterization of model companionship.

Note that $T_{\forall \vee \exists }^\tau $ may convey more information than $T_{\forall }^\tau \cup T_{\exists }^\tau $ , as there could be $\psi \vee \phi $ (with $\psi $ universal and $\phi $ existential) which is in the former but does not follow from the latter.

Note that a model complete $\tau $ -theory T is the model companion of $T^\tau _\forall $ and the AMC of $T^\tau _{\forall \vee \exists }$ .

A $\tau $ -theory S can have an AMC if, to some extent, it has already maximized the family of existential sentences which are consistent with its universal consequences. Let us elaborate more on this point. Given a $\tau $ -theory T, the family of strongly $T_{\forall \vee \exists }^\tau $ -consistent $\Pi _2$ -sentences for $\tau $ describes a fragment of the $\Pi _2$ -sentences which hold in a T-ec $\tau $ -structure (see [Reference Viale41, fact 2.2.5]). An AMC describes the set-up in which the choice of $\tau $ is such that the family of strongly $T_{\forall \vee \exists }^\tau $ -consistent $\Pi _2$ -sentences axiomatizes a model complete theory. This leads to another peculiar aspect separating model-companionship from absolute model companionship:

On the contrary this property fails for model companionship: the universal fragment of the theory of $(\mathbb {Q},+,\cdot ,0,1)$ is inconsistent with the $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -theory of algebraically closed fields. As we already mentioned, complete first order $\tau $ -theories are model companionable if and only if they admit an AMC. Indeed, for a complete theory S, the information conveyed by $S^\tau _\forall $ and by $S^\tau _{\forall \vee \exists }$ is the same. On the other hand we have already seen that there are non-complete $\tau $ -theories admitting a model companion but not an AMC, e.g., the $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -theory of fields. Let us briefly analyze more in details what is the obstruction for the $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -theory of fields of characteristic $0$ to have the $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -theory of algebraically closed fields of characteristic $0$ as its AMC. The problem is that Diophantine equations can be expressed by atomic formulae of $\left \lbrace +,\cdot ,0,1 \right \rbrace $ and not all of them have rational solutions. On the other hand the existence of solutions for these equations does not contradict the field axioms. Note that the smallest existentially closed $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -model $\mathcal {M}$ for the theory of fields is given by the algebraically closed field consisting exactly of the solutions of diophantine equations. Its standard construction builds it as a direct limit of Galois extensions of the rationals each adding new solutions to polynomial equations with coefficients in $\mathbb {N}$ but without rational solutions. By performing this construction we can preserve the field axioms and the characteristic $0$ (which holds in $\mathbb {Q}$ as well as in all the Galois extensions of $\mathbb {Q}$ under consideration), but at each stage we may invalidate some universal sentence that is true in $\mathbb {Q}$ (i.e., a universal sentence asserting that a certain diophantine equation does not have rational solutions). In particular, the closing-off process performed in signature $\sigma =\left \lbrace +,\cdot ,0,1 \right \rbrace $ which brings from the theory of fields T to that of algebraically closed fields cannot be described only by the $\Pi _2$ -sentences $\psi $ which are strongly $T_{\forall \vee \exists }^\tau $ -consistent for $\sigma $ . Indeed, there are other $\Pi _2$ -sentences which need to be realized in an algebraically closed field and which are not strongly $T_{\forall \vee \exists }^\tau $ -consistent for $\sigma $ (i.e., the existential statements asserting the existence of solutions for diophantine equations). In particular, the $\left \lbrace +,\cdot ,0,1 \right \rbrace $ -theory of algebraically closed fields is the AMC only of the theory of integral domains which extends the field of algebraic numbers (i.e., which have solutions to all polynomial equations expressible in the signature $\left \lbrace +,\cdot ,0,1 \right \rbrace $ ).

On the other hand we will argue that in the case of set theory, AMC faithfully describes the closing-off process of models of set theory with respect to basic set-theoretic operations.

5.3 The (A)MC-spectra of a first order theory

Model theory has been extremely successful in classifying the complexity of a mathematical theory according to its structural properties, and has produced a variety of properties and criteria to separate the (so-called) tame mathematical theories from the (so-called) untame ones (e.g., o-minimality, stability, simplicity, NIP), see [Reference Baldwin8] for a philosophical appraisal of this theme. Typically a mathematical theory is considered hard to classify (and thus untame) if it can code in itself first order arithmetic. In this respect the $\in $ -theory $\mathsf {ZFC}$ is untame.

We have already observed that most mathematical theories admit many different first order axiomatizations in (almost) as many distinct signatures. A common characteristic of tameness properties such as o-minimality, stability, simplicity, and NIP is that they are signature invariant. More precisely, if we take a $\tau $ -theory T and we consider its Morleyization $T^*$ in the signature $\tau ^*$ (see Notation 5.1), T is o-minimal (stable, simple, and NIP) if and only if so is $T^*$ . In contrast, this is not the case for Robinson’s notion of model companionship, as we have already observed for the theory of integral domains. We instantiate the model-theoretic criterion to select signatures formulated in the last part of Section 4 by means of the following:

Clearly $\mathfrak {spec}_{\mathsf {MC}}(T,\tau )$ is a superset of $\mathfrak {spec}_{\mathsf {AMC}}\left (T,\tau \right )$ and the two can be distinct. Observe that $C=\mathsf {Form}_\tau \times \left \lbrace 0 \right \rbrace $ is always in the AMC-spectrum of a theory T, as $T+T_{\tau ,C}$ admits quantifier elimination and therefore it is model complete and its own AMC in signature $\tau _C=\tau ^*$ . Moreover, $\emptyset $ is in the (A)MC-spectrum of T if and only if T has an (absolute) model companion. For $\sigma =\left \lbrace +,\cdot ,0,1 \right \rbrace $ and the various $\sigma $ -theories considered in Section 3 we have that:

• • $\emptyset $ is in $\mathfrak {spec}_{\mathsf {MC}}(T,\sigma )\setminus \mathfrak {spec}_{\mathsf {AMC}}(T,\sigma )$ for T the $\sigma $ -theory of commutative (semi)rings with no $0$ -divisors or the theory of fields.

• • For $\phi $ the $\sigma $ -formula defining the additive inverse, $\left \lbrace (\phi ,1) \right \rbrace $ is in $\mathfrak {spec}_{\mathsf {MC}}(T,\sigma )$ if T is the theory of commutative rings with no $0$ -divisors or the theory of fields, while $\left \lbrace (\phi ,1) \right \rbrace $ is not in $\mathfrak {spec}_{\mathsf {MC}}(S,\sigma )$ if S is the $\sigma $ -theory of commutative semirings with no $0$ -divisors.

• • For $\psi $ the $\sigma $ -formula defining the multiplicative inverse, $\left \lbrace (\psi ,1) \right \rbrace $ is in $\mathfrak {spec}_{\mathsf {MC}}(T,\sigma )$ if T is the theory of fields, while $\left \lbrace (\psi ,1) \right \rbrace $ is not in $\mathfrak {spec}_{\mathsf {MC}}(S,\sigma )$ if S is the $\sigma $ -theory of commutative (semi)rings with no $0$ -divisors.

• • It can be shown that $\mathfrak {spec}_{\mathsf {AMC}}(T,\sigma )=\emptyset $ for T the $\sigma $ -theory of commutative (semi)rings with no $0$ -divisors or the theory of fields.

A category theoretic perspective on the notion of (A)MC spectrum can again be enlightening: given a $\tau $ -theory T, consider the category $\mathcal {C}_T$ given by the elementary class of its $\tau $ -models as objects and the $\tau $ -morphisms (e.g., maps which preserve the atomic $\tau $ -formulae) between them as arrows. By taking a subset A of $\mathsf {Form}_\tau \times 2$ we can pass to the category $\mathcal {C}_{T+T_{\tau ,A}}$ whose objects are $\tau _A$ -models of $T+T_{\tau ,A}$ and whose arrows are the $\tau _A$ -morphisms (in accordance with Notation 5.1). There is a natural identification of the objects of $\mathcal {C}_T$ and those of $\mathcal {C}_{T+T_{\tau ,A}}$ , but the arrows of $\mathcal {C}_{T+T_{\tau ,A}}$ are now a possibly much narrower subfamily of the arrows of $\mathcal {C}_{T}$ . Important structural information on a $\tau $ -theory T is given (at least from a categorial point of view) by classifying which sets A produce an elementary class of $\tau _A$ -models for which the $T+T_{\tau ,A}$ -ec models constitute themselves an elementary class. The model companionship spectrum of T gives exactly this structural information on the arrows of $\mathcal {C}_T$ . Similar considerations apply for the AMC-spectrum of a theory. On the other hand simplicity, stability, and NIP provide fundamental structural informations on the class of objects of $\mathcal {C}_T$ , but it is not transparent whether they also convey information on its class of arrows.

6.1 What is the right signature for set theory?

The standard axioms of $\mathsf {ZFC}$ in the $\in $ -signature are clearly sufficient to provide a first order axiomatization of set theory. However a closer inspection reveals that many simple set-theoretic concepts are not formalized by simple $\in $ -formulae.

Consider, for example, the notion of ordered pair. While we informally write $x=\langle y,z \rangle $ to mean that x is the ordered pair with first component y and second component z, in set theoretic terms this statement hides a non-trivial coding of the concept of ordered pair (for example) by means of Kuratowski’s definition: $x=\left \lbrace \left \lbrace y \right \rbrace ,\left \lbrace y,z \right \rbrace \right \rbrace $ . A proper definition of the concept of ordered pair in the $\in $ -signature can then be given by the following $\in $ -formula:

$$\begin{align*}&\exists t\exists u\;[\forall w\,(w\in x\leftrightarrow w=t\vee w=u)\\&\quad \wedge\forall v\,(v\in t\leftrightarrow v=y) \wedge\forall v\,(v\in u\leftrightarrow v=y\vee v=z)]. \end{align*}$$

It is clear that the meaning of this $\in $ -formula is hardly recognizable with a rapid glance (unlike $x=\langle y,z \rangle $ ). Moreover, from a purely logical perspective, its Lévy complexity is already $\Sigma _2$ . This clashes with our understanding that the concept of ordered pair is simple. Indeed, we do not regard the notion of ordered pair as a complex concept, contrary to other more complicated and theoretically loaded ones like that of uncountability, or many of the properties of the continuum (such as its correct place in the hierarchy of uncountable cardinals). In a similar vein other very basic notions such as being a function, a binary relation, or the domain or the range of a function are formalized by rather complicated $\in $ -formulae, both from the point of view of their readability and of their Lévy complexity.

The standard solution adopted in set theory textbooksFootnote ³⁰ is to regard as basic all those $\in $ -formulae in which the quantifiers are bounded to range over the elements of some set, that is the $\Delta _0$ -formulae. In order to make these observations precise we need to be extremely cautious on our notational conventions.

Notation 6.1 For any $A\subseteq \mathsf {Form}_{\left \lbrace \in \right \rbrace }\times 2$ we write $\in _A$ rather than $\left \lbrace \in \right \rbrace _A$ , and we let $T_{\in ,A}$ be the $\in _A$ -theory

$$\begin{align*}T_{\left\lbrace \in \right\rbrace,A}+\forall x \,\left[ (\forall y\,y\notin x)\leftrightarrow c_{\left\lbrace \in \right\rbrace}=x \right], \end{align*}$$

where the theory $T_{\left \lbrace \in \right \rbrace ,A}$ (according to Notation 5.1 for $\left \lbrace \in \right \rbrace $ and A) is reinforced by an axiom asserting that the interpretation of the constant symbol $c_{\left \lbrace \in \right \rbrace }$ is the empty set. We use the abbreviations $\in _{\Delta _0}$ and $T_{\Delta _0}$ to denote what, according to the above conventions, should rather be slight extensions ofFootnote ³¹ $\in _{\Delta _0\times \left \lbrace 0 \right \rbrace }$ and $T_{\in ,\Delta _0\times \left \lbrace 0 \right \rbrace }$ .

For any $\tau \supseteq \in _{\Delta _0}$ we also let $\mathsf {ZFC}_\tau $ denote $\mathsf {ZFC}+T_{\Delta _0}$ enriched with the replacement axiomFootnote ³² for all $\tau $ -formulae, and $\mathsf {ZFC}^-_\tau $ denote $\mathsf {ZFC}_\tau $ without the powerset axiom. $\mathsf {ZFC}_{\Delta _0}$ denotes $\mathsf {ZFC}_\tau $ for $\tau $ being $\in _{\Delta _0}$ , and similarly we define $\mathsf {ZFC}^-_{\Delta _0}$ .

The reason why set-theoretic practice regards the concepts expressed by quantifier-free formulae of $\in _{\Delta _0}$ as “simple” is the fact that the truth value of these formulae is invariant among transitive models of large enough fragments of $\mathsf {ZF}$ (e.g., [Reference Kunen25, corollary IV.3.6]), and thus also forcing invariant (e.g., [Reference Jech22, lemma 14.21]). Furthermore $\in _{\Delta _0}$ is a signature which allows one to formalize many fundamental set theoretic concepts using formulae whose Lévy complexity is in accordance to our intuitive understanding (e.g., [Reference Jech22, chap. 13, lemma 13.10]).

For example consider the following notions and their logical complexity:

• • $(x$ is a cardinal $)$ is the $\Pi _1$ -formula (for $\in _{\Delta _0}$ )

  

• • $(x$ is $\aleph _1)$ is the boolean combination of $\Sigma _1$ -formulae

  

• • ${\mathsf {CH}}$ is the $\Sigma _2$ -sentence

  
  and $\neg {\mathsf {CH}}$ is the boolean combination of $\Pi _2$ -sentencesFootnote ³³
  

• • $(x\ {is}\ \aleph _2)$ is the $\Sigma _2$ -formula

  $$ \begin{align*} \hspace{255pt}(x\text{ is a cardinal})\wedge \end{align*} $$
  $\hspace{-30pt}\wedge\exists F\exists y\,\left[ (y\ {is}\ \aleph_1)\wedge (y\in x)\wedge(F:y\times x\to x) \wedge \forall\alpha\in x\,(F\restriction y\times\left\lbrace \alpha \right\rbrace\text{ is a surjection on }\alpha) \right]. $

• • $2^{\aleph _0}>\aleph _2$ is the boolean combination of $\Pi _2$ -sentences

  

• • $2^{\aleph _0}\leq \aleph _2$ is the $\Sigma _2$ -sentence

  

Let us also introduce the notation we will use to handle the substructure relation over expanded signatures. The following conventions supplement Notation 5.1.

6.2 Lévy absoluteness and the possible AMCs of set theory

Recall that for an infinite cardinal $\lambda $ , $H_{\lambda }$ is the initial transitive fragment of V given by those sets whose transitive closure has size less than $\lambda $ and is by itself a set in V (see [Reference Kunen25, sec. IV.6]). We have the following results.

• Stratification of V: The universe of sets V can be stratified as the union, along the class of infinite cardinals $\kappa $ , of the sets $H_{\kappa ^+}$ .

• Standard theory of $H_{\lambda }$ : [Reference Kunen25, theorem IV.6.5] For any uncountable cardinal $\lambda $ , $(H_{\lambda },\in _{\Delta _0}^V)$ is a model of $\mathsf {ZFC}^-_{\Delta _0}$ (recall Notation 6.1).

• Strong Lévy absoluteness: [Reference Venturi and Viale38, lemma 5.3] For all $A_i\subseteq \mathcal {P}\left (\kappa \right )^{n_i}$ for $i=1,\dots ,k$ ,

  $$\begin{align*}(H_{\kappa^+}^V,\in_{\Delta_0}^V,A_1,\dots,A_k)\prec_1(V,\in_{\Delta_0}^V,A_1,\dots,A_k). \end{align*}$$

• Second order characterization of $H_{\kappa ^+}$ : Whenever $\mathcal {M}$ is an $\in _{\Delta _0}\cup \left \lbrace \kappa \right \rbrace $ -model of $\mathsf {ZFC}_{\Delta _0}+\kappa $ is an infinite cardinal $(H_{\kappa ^+}^{\mathcal {M}},\in _{\Delta _0}^{\mathcal {M}},\kappa ^{\mathcal {M}})$ is a model of the $\Pi _2$ -sentence for $\in _{\Delta _0}\cup \left \lbrace \kappa \right \rbrace $

  (7) $$ \begin{align} \forall x\exists f\,(f:\kappa\to x\text{ is a surjection}). \end{align} $$

  Furthermore $H_{\kappa ^+}^V$ can be described as the unique set $M\in V$ such that:

  • • $\kappa \in M$ and M is transitive;

  • • $(M,\in _{\Delta _0}^V)$ satisfies (7) and $\mathsf {ZFC}^-_{\Delta _0}$ ;

  • • M has the (second order property) that for all $a\in M$ and $b\subseteq a$ , $b\in M$ as well.

In particular, the theory of each $H_{\kappa ^+}$ offers a $\Sigma _1$ -approximation of the theory of V with respect to a signatures $\tau _\kappa $ which extends $\in _{\Delta _0}\cup \left \lbrace \kappa \right \rbrace $ and contains predicates for subsets of $\mathcal {P}\left (\kappa \right )^n$ for some $n\in \mathbb {N}$ . Consequently, if we want to study the $\Sigma _1$ -properties of the reals (second order arithmetic), or the powerset of the reals (third order arithmetic) it is sufficient to study them within the theory of some $H_{\kappa ^+}$ for $\kappa $ a large enough regular cardinal (and for most purposes $H_{\aleph _1}$ suffices for second order arithmetic, and $H_{\aleph _2}$ for third-order arithmetic). Moreover, by Lévy absoluteness, $H_{\kappa ^+}$ and V agree on the computation of many simple set theoretic operations and relations: for example the operations and relations formalizable by means of $\Delta _0$ -formulae and $\Delta _0$ -definable Skolem functions.

The above results entail that whenever T is a $\tau $ -theory that extends $\mathsf {ZFC}_{\Delta _0}+\kappa $ is a cardinal, $\tau $ is a signature that contains $\in _{\Delta _0}\cup \left \lbrace \kappa \right \rbrace $ and is contained in

$$\begin{align*}\in_{\Delta_0}\cup\left\lbrace A: A\subseteq\mathcal{P}\left(\kappa\right)^n,\, n\in\mathbb{N} \right\rbrace, \end{align*}$$

and $(V,\tau ^V)$ models T, then

$$\begin{align*}(H_{\kappa^+}^V,\tau^V)\prec_1 (V,\tau^V). \end{align*}$$

Hence $(H_{\kappa ^+},\tau ^V)$ witnesses that (7) is consistent with the universal fragment of the $\tau $ -theory of V. In particular, if T has an AMC for $\tau $ , say S, then (7) must be in S, since it is a $\Pi _2$ -sentence for $\tau $ which is strongly $T^\tau _{\forall \vee \exists }$ -consistent.

We can now collect all these observations and conclude the following.

1. (1) The structure of V is uniquely determined by the structure of the various $H_{\kappa ^+}$ as $\kappa $ ranges among the infinite cardinals.

2. (2) Given an infinite cardinal $\kappa $ , consider a signature $\tau $ extending $\in _{\Delta _0}$ only with predicates for subsets of $\mathcal {P}\left (\kappa \right )$ and a constant symbol for $\kappa $ . Then, if some ${T\supseteq \mathsf {ZFC}_\tau }$ has an AMC, this AMC looks like a theory of $H_{\kappa ^+}$ and is axiomatized by the $\Pi _2$ -sentences for $\tau $ which are strongly $T^\tau _{\forall \vee \exists }$ -consistent.

Now assume T is some $\in $ -theory $T\supseteq \mathsf {ZFC}+$ large cardinals and $A\subseteq \mathsf {Form}_\in \times 2$ is in $\mathfrak {spec}_{\mathsf {AMC}}\left (T,\left \lbrace \in \right \rbrace \right )$ . Assume further that:

• • For some constant symbol $\kappa $ of $\in _A$

  $$\begin{align*}\mathsf{ZFC}^-+\forall X\exists f\,(f:\kappa\to X\text{ is a surjection}) \end{align*}$$
  is in $\mathsf {AMC}(T,A)$ .

• • Every atomic formula of $\in _A$ satisfies at least one of the simplicity criteria set forth in (a).

Then (on the basis of (IEC)) we should conclude that $\mathsf {AMC}(T,A)$ describes the correct theory of $H_{\kappa ^+}$ , since criterion (B) is satisfied by all axioms of $\mathsf {AMC}(T,A)$ .

6.3 Existentially closed fragments of the set theoretic universe versus AMC

We conclude this section by giving a non-exhaustive list of results showing that, for certain infinite cardinals $\kappa $ , the corresponding $H_{\kappa ^+}$ already witnesses some important existential closure properties (e.g., Lévy absoluteness, Shoenfield’s absoluteness, $\mathsf {BMM}$ , and $\mathsf {BMM}^{++}$ ) and its theory has some traits which are peculiar of model complete theories (e.g., Woodin’s absoluteness, ${\mathsf {MM}}^{+++}$ ). The reader does not need to be familiar with these results, as they only serve as motivation to bring our focus on existentially closed models for set theory.Footnote ³⁴

• Lévy absoluteness: [Reference Venturi and Viale38, lemma 5.3] Whenever $\lambda $ is a regular uncountable cardinal,

  $$\begin{align*}(H_\lambda,\in_{\Delta_0}^V)\prec_1 (V,\in_{\Delta_0}^V). \end{align*}$$

• Shoenfield’s absoluteness: (see [Reference Viale40, lemma 1.2] for the apparently weaker formulation we give here.) Whenever G is V-generic for some forcing notion in V,

  

• Woodin’s absoluteness: (see [Reference Viale40, lemma 3.2] for the weak form of Woodin’s result we give here.) Whenever G is V-generic for some forcing notion in V (and there are class many Woodin cardinals in V),

  

• Bounded Martin’s Maximum: ( $\mathsf {BMM}$ ) [Reference Bagaria6] Whenever G is V-generic for some stationary set preserving forcing notion in V,

  

• $\mathsf {BMM}^{++}$ : [Reference Woodin47, definition 10.91] Whenever G is V-generic for some stationary set preserving forcing notion in V,

  
  where is a unary predicate symbol interpreted by the non-stationary ideal on .

• Bounded category forcing axioms, ${\mathsf {MM}}^{+++}$ , [Reference Asperó and Viale4, Reference Audrito and Viale5, Reference Viale39] Whenever V and $V[G]$ are models of ${\mathsf {MM}}^{+++}$ ( , $\mathsf {BCFA}({\mathsf {SSP}})$ ) and G is V-generic for some stationary set preserving forcing notion in V,

  

• Woodin’s axiom: $(*)$ (see [Reference Larson26, definition 7.5], or [Reference Asperó and Schindler3] for the formulation given here.) Whenever there are class many Woodin cardinals, is a precipitous ideal, and G is V-generic for some stationary set preserving forcing notion in V

  
  (where $A^{V[G]}$ is the canonical interpretation in $L(\mathbb {R})^{V[G]}$ of the definition of A in $L(\mathbb {R})$ ).

7.1 The AMCs of set theory are theories of $H_{\kappa ^+}$

The first result shows that the AMC spectrum of set theory isolates a rich set of theories which produce models of $\mathsf {ZFC}^-$ , that is, structures which behave like an $H_{\lambda }$ for some regular uncountable cardinal $\lambda $ .

7.2 Forcibility versus absolute model companionship

The following is the major result relating AMC to forcibility and to forcing axioms:Footnote ³⁷

Theorem 7.4 Let S be the $\in $ -theory

$$\begin{align*}\mathsf{ZFC}+ {\mathrm{there\ exist\ class\ many\ supercompact\ cardinals}}. \end{align*}$$

Then there is a recursive setFootnote ³⁸ $B\in \mathfrak {spec}_{\mathsf {AMC}}\left (S,\in \right )$ with $\in _B$ containing $\in _{\Delta _0}$ , and such that for any $\Pi _2$ -sentence $\psi $ for $\in _B$ and any $\in $ -theory $R\supseteq S$ the following are equivalent:

1. (a) $\psi \in \mathsf {AMC}(R,B)$ ;

2. (b) $(R+T_{\in ,B})^{\in _B}_{\forall \vee \exists }+S+{\mathsf {MM}}^{++}+T_{\in ,B}$ provesFootnote ³⁹ ;

3. (c) R proves that is forcibleFootnote ⁴⁰ by a stationary set preserving forcing;

4. (d) R proves that is forcible by some forcing;

5. (e) For any $R'\supseteq R$ , $\psi +(R'+T_{\in ,B})^{\in _B}_{\forall \vee \exists }$ is consistent.

Furthermore for any $\theta $ which is a boolean combination of $\Pi _1$ -sentences for $\in _B$ and any $(V,\in )$ model of S, TFAE:

1. (A) $(V,\in _B^V)$ models $\theta $ ;

2. (B) $(V,\in _B^V)$ models that some forcing notion P forces $\theta $ ;

3. (C) $(V,\in _B^V)$ models that all forcing notions P force $\theta $ .

The equivalence of (a) and (b) shows that the $\in _B$ -theory of $H_{\aleph _2}$ is model complete in models of strong forcing axioms; we give a partial converse in Theorem 7.9.

The second part of the theorem shows that forcing cannot change the $\Pi _1$ -fragment of the theory of V in signature ${\in _B}\supseteq {\in _{\Delta _0}}$ . Moreover, a direct consequence of the above result is that if $(V,\in )$ is a model of S and $S^*$ is the $\in _B$ -theory of its unique extension to a model of $T_{\in ,B}$ , we get that a $\Pi _2$ -sentence $\psi $ for $\in _B$ is consistent with the universal fragment of $S^*$ if and only if is forcible over V.

7.3 The AMC-spectrum of set theory and the continuum problem

The study of the AMC spectrum of set theory places $\aleph _2$ in a very special position among the possible values of the continuum.

Let us briefly argue why ${\mathsf {CH}}$ cannot be in $\mathsf {AMC}(R,A)$ whenever $R+\neg {\mathsf {CH}}$ is consistent.

Let us also argue why $\neg {\mathsf {CH}}$ falls in the AMC of $R+T_{\in ,B}$ for the set $B\subseteq \mathsf {Form}_\in \times 2$ and the theory R of (2) above. Towards this aim we appeal to the equivalence between (A) and (C) of Theorem 7.4 (which gives a precise instantiation of simplicity criterion (iii)). First of all notice that an $\in $ -theory R is complete if and only if so is $R+T_{\in ,B}$ . Let $R_0$ be some $\in $ -completion of $R\supseteq S$ and $(V,\in )$ a model of $R_0$ . We now note that $\in _B$ has a constant $\kappa $ to denote

, $\mathsf {AMC}(S,B)$ satisfies the $\in _{\Delta _0}\cup \left \lbrace \kappa \right \rbrace $ -sentence “ $\kappa $ is the first uncountable cardinal”, $\mathsf {AMC}(R_0,B)=\mathsf {AMC}(S,B)+(R_0+T_{\in ,B})_{\forall \vee \exists }$ (by Lemma 5.4). Let $\mathsf {B}$ be a cba such that

holds in $(V,\in )$ .Footnote ⁴² Let G be any ultrafilter on $\mathsf {B}$ . Then

(by [Reference Jech22, lemma 14.14]) and $V^{\mathsf {B}}/_G$ also models the universal and existential $\in _B$ -fragment of $R+T_{\in ,B}$ (by (A) $\Leftrightarrow $ (C) of Theorem 7.4, and [Reference Jech22, lemma 14.14]). In particular, the $\Pi _2$ -conjunct of $\neg {\mathsf {CH}}$ is consistent with the universal and existential fragment of any completion of $S+T_{\in ,B}$ , hence it belongs to the AMC of $S+T_{\in ,B}$ , while the $\Sigma _2$ -conjunct of $\neg {\mathsf {CH}}$ is in the AMC of $S+T_{\in ,B}$ since the axiom $\mathsf {Ax}^1_\psi $ for $\psi $ defining the first uncountable cardinal is in $T_{\in ,B}$ .

In particular, we see that forcing becomes a powerful tool to prove that a $\Pi _2$ -sentence formalizable in $\in _B$ is in the AMC of $S+T_{\in ,B}$ . Indeed, it suffices to prove over S that this sentence is forcible. More generally minimal variants of the above argument give:

This shows that the argument described for ${\mathsf {CH}}$ is far more general and can be used to single out many other set theoretic properties. We can apply it for example to $2^{\aleph _0}>\aleph _2$ : Moore introduced in [Reference Moore32] a $\Pi _2$ -sentence $\theta _{\mathrm {Moore}}$ for $\in _{\Delta _0}$ to show the existence of a definable well order of the reals in type in models of the bounded proper forcing axiom.Footnote ⁴³

The two theorems single out $2^{\aleph _0}=\aleph _2$ among all possible solutions of the continuum problem.Footnote ⁴⁶ For any $\in $ -theory R extending $\mathsf {ZFC}+$ large cardinals there is at least one $B\in \mathfrak {spec}_{\mathsf {AMC}}\left (R,\in \right )$ with $\neg {\mathsf {CH}}$ (actually with a definable version of $2^{\aleph _0}=\aleph _2$ ) in $\mathsf {AMC}(R,B)$ , and this occurs even if $R\models {\mathsf {CH}}$ or $R\models 2^{\aleph _0}>\aleph _2$ . On the other hand for any $\in $ -theory R extending $\mathsf {ZFC}$ , if ${\mathsf {CH}}$ is independent of R, then ${\mathsf {CH}}$ is never in $\mathsf {AMC}(R,A)$ for any $A\in \mathfrak {spec}_{\mathsf {AMC}}\left (R,\in \right )$ (if $\in _A$ contains $\in _{\Delta _0}$ ) and similarly if $\theta _{\mathrm {Moore}}$ is independent of R, $2^{\aleph _0}>\aleph _2$ is never in $\mathsf {AMC}(R,A)$ for any $A\in \mathfrak {spec}_{\mathsf {AMC}}\left (R,\in \right )$ (if $\in _A$ contains $\in _{\Delta _0}$ ).Footnote ⁴⁷

7.4 Model completeness for the theory of $H_{\aleph _2}$ versus bounded forcing axioms and Woodin’s axiom $(*)$

Finally we can show that for certain signatures $\tau $ and a model $(V,\in )$ of $\mathsf {ZFC}+$ large cardinals, a model completeness result for the $\tau $ -theory of $H_{\aleph _2}^V$ is equivalent to assert that V satisfies a strong form of Woodin’s axiom $(*)$ . The reader interested in the (so far) optimal result is referred to [Reference Viale41, theorem 5.6]. Recall the axioms ${\mathsf {BMM}}$ , ${\mathsf {BMM}}^{++}$ , $(*)$ presented in Section 6.3. An immediate consequence of the definition of $\in _B$ and of the invariance of the universal $\in _B$ -theory of V across forcing extensions is the following result.Footnote ⁴⁸

Note that

is a precipitous ideal in V, if V models Martin’s maximum (of which ${\mathsf {BMM}}$ is a weakening). The theorem follows almost immediately from Theorem 7.4 by the observation that if G is V-generic for a stationary set preserving forcing and $(V,\in )$ models S

8.1 Maximality

Maximality principles play an important role in the study of the axiomatic extensions of $\mathsf {ZFC}$ . Maximality conveys the idea that the universe of sets is as full as it could be, while maximality principles are (first order) statements which try to capture this idea axiomatically.Footnote ⁴⁹

There is no unique way to realize maximality in set theory, and its various forms depend on what in fact is maximized. Indeed, the process of maximization can be applied to objects, or possibilities.Footnote ⁵⁰

• • With respect to objects we find two standard forms of maximization: height maximality (expressing maximality for the lengths of the series of ordinals) and width maximality (expressing the maximality of the power-set operation). The former is successfully exemplified by large cardinals axioms [Reference Koellner24], while the latter represents a cluster of notions often connected to the justification of forcing axioms, generic absoluteness principles [Reference Bagaria7], and forms of horizontal reflection like the Inner Model Hypothesis [Reference Antos, Barton and Friedman1].

• • For what concerns the maximization of possibilities, we find modal principles of forcing like the so-called Maximality Principle [Reference Hamkins18] (expressing the idea that everything that is possibly necessary is true) or resurrection axioms [Reference Asperó and Viale4, Reference Audrito and Viale5, Reference Hamkins and Johnstone20] (expressing the idea that if something is true, then it is necessarily possible). This form of maximality is closely related to forms of generic absoluteness, but it differs from it conceptually. Indeed, the justification of these modal principles often relays on potentialist views of set theory [27, Reference Shapiro28] that are not always compatible with the actualist ones that motivate generic absoluteness principles.Footnote ⁵¹

Now, what form of maximality is displayed by the notion of model companionship? Can it be compared to any of the above forms of set-theoretical maximality?

A first step toward answering the above questions consists in analyzing whether the maximality provided by models companions concerns primarily the underlying ontology (thus suggesting a form of maximization of objects), or the class of sentences that are considered valid (as it seems the case when considering generic absoluteness principles or modal principles of forcing). In other words, does the maximality provided by model companions concerns primarily the ontology of sets, or the family of set-theoretic truths? This question does not have a clear-cut answer. As a matter of fact, we believe that model companionship results for set theory display a multifaceted character related to both forms of maximality.

For what concerns height maximality, this notion is successfully realized by large cardinal axioms. While the model companionship results for set theory do not seem to enforce this form of maximality, they can nonetheless be used to justify it (we argue on this point in the next section). On the other hand, for what concerns width maximality, notice that bounded forcing axioms (and similar principles meant to express forms of horizontal maximality) can be reformulated in terms of generic absoluteness properties for . Many of the results we presented in the previous section outlines that such generic absoluteness results come in pairs with the model completeness for the theory of $H_{\aleph _2}$ with respect to natural signatures for this structure (e.g., Theorem 7.9 or [Reference Viale41, theorem 5.6]). Note that the standard formulation of bounded forcing axioms, such as ${\mathsf {BMM}},{\mathsf {BMM}}^{++},(*)$ , yield forms of absoluteness for the theory of $H_{\aleph _2}$ , which are (apparently) weaker than the model completeness result given by Theorem 7.4: they only assert that (for the appropriate fragment $\tau $ of $\in _B$ ) $H_{\aleph _2}$ is $\Sigma _1$ -elementary for $\tau $ only with respect to certain forcing extensions of V (which are $\tau $ -models of the universal $\tau $ -theory of V by the second part of Theorem 7.4). On the contrary, model completeness of the $\in _B$ -theory of $H_{\aleph _2}$ yields $\Sigma _1$ -elementarity for the much larger class of all $\in _B$ -superstructures which satisfy the same universal $\in _B$ -theory.

For what concerns the modal principles of forcing, we can argue as for width maximality. As a matter of fact, the Maximality Principle and resurrection axioms, as exposed in [Reference Asperó and Viale4, Reference Audrito and Viale5, Reference Hamkins18, Reference Hamkins and Johnstone20] are, by all means, principles of generic absoluteness for the theory of $H_{\aleph _2}$ (at least in the presence of mild forms of bounded forcing axioms). Indeed, for many of these principles their natural formulation asserts that the theory of $H_{\aleph _2}$ is given by those $\Pi _2$ -sentences $\varphi $ (for an appropriate fragment of $\in _B$ ) for which $\varphi \to \square \Diamond \varphi $ holds in the (Kripke frame given by the) generic multiverse composed of initial segments of forcing extensions of V. Therefore, they describe the theory of $H_{\aleph _2}$ with respect to the generic multiverse much like infinitely generic structures (introduced by Robinson in connection with the notion of infinite forcing) describe the theory of the existentially closed models for a first order theory. Leveraging on the model completeness results for $\in _B$ we can actually identify the theory of $H_{\aleph _2}$ in models of strong forcing axioms with that given by the infinitely generic structures, in signature $\in _B$ , for the class of models of $\mathsf {ZFC}+$ large cardinals. Hence the absoluteness displayed by the model companions of set theory, relative to $\in _B$ , largely encompasses that of the modal principles of forcing, since the latter are infinitely generic (at least on the face of their definition) only with respect to the generic multiverse and not with respect to the elementary class of the $\in _B$ -theory of $\mathsf {ZFC}_B+$ large cardinals.Footnote ⁵²

Finally, what is the connection between the maximization given by model companionship results and the (somewhat vague) form of completeness that is encoded in the principle that we called Hilbertian Completeness? Although our aim, similar to Hilbert’s, is to implement the idea that the universe $(V, \in )$ contains all possible sets, our formal way to make this idea concrete differs significantly from Hilbert’s approach. Instead of focusing directly on the objects, we maximize existential sentences using our principle of Informal Existential Completeness (IEC). Thus, maximality of the objects receives in our approach a syntactic twist, being formulated in terms of maximality of solutions to simple problems (according to a syntactic measure of simplicity given by the selection of a specific first order signature).

Summing up, it seems that the form of maximality provided by (absolute) model companionship does not precisely fit with any of the standard forms of set-theoretical maximality. Nonetheless there are very strong similarities with generic absoluteness principles and certain types of forcing axioms,Footnote ⁵³ and conceptual similarities with Hilbert’s idea of completeness. We see the notion of model companionship as a useful tool towards a mathematical formalization of the somewhat vague ideas of width maximality and Hilbertian completeness. Our opinion is that the model companions of set theory yield a form of maximality that we may call algebraic maximality. The inspiration clearly comes from algebraically closed fields. We consider algebraic maximization of a theory T as a two steps process: first, one determines which are the basic, simple, concepts of the theory of interest; then, once the language is extended to a signature $\tau $ with symbols for these concepts, one analyzes the T-ec $\tau $ -models. T is algebraically maximal for $\tau $ in case the $\tau $ -theory of the T-ec $\tau $ -models is model complete and also when its T-ec $\tau $ -models can be seen as “parts of” the models of T. This can be formally implemented as follows:

In terms of algebraic maximality we briefly review the results of Section 7. First of all notice that Theorem 7.3 states that for any extension T of $\mathsf {ZFC}$ , algebraic maximality can be obtained for T by sets $A_\kappa $ such that the AMC of T in signature $\in _{A_\kappa }$ describes the theory of $H_{\kappa ^+}$ in models of T, as $\kappa $ ranges among the T-definable cardinals.Footnote ⁵⁵ Moreover, the first part of Theorem 7.4 states that algebraic maximality, relative to B, holds for the theory $\mathsf {ZFC}+{large cardinals} +$ forcing axioms, while its second part states that $\in _B$ satisfies the simplicity criteria set forth in (IEC) of Section 6. Finally, we have that Theorem 7.9 (and its optimal strengthening given by [Reference Viale41, theorem 5.6]) shows that algebraic maximality for the $\in _B$ of Theorem 7.4 entails (or is equivalent to) certain strong forcing axioms.

In conclusion we can argue that Theorems 7.4 and 7.9 show that algebraic maximality for set theory, relative to $\in _B$ , describes the correct axioms for the theory of $H_{\aleph _2}$ , as algebraic maximality for $\in _B$ realizes (IEC) for this fragment of the universe. This naturally lead us to discuss the second topic of this section: justification.

8.2 Justification

Let us now turn to the topic of justification. When it comes to justifying new set-theoretical principles, the proposed reasons normally fall into two main categories: intrinsic or extrinsic. These forms of justification have been developed within the so-called Gödel’s program:Footnote ⁵⁶ a step-by-step extension of $\mathsf {ZFC}$ , aimed to coherently complete our picture of the universe of sets. In this sense, the present approach is clearly in the wake of Gödel’s program. Indeed, the axiomatization of the model companions of set theory should, eventually, provide a description of the theory of V as the stratification of the theories of the various $H_{\kappa ^+}$ . Furthermore, the essential role that large cardinals and strong forms of forcing axioms play in the individuation of these model companionship results suggests a justification of the truth of these axioms in view of the nice model-theoretic properties for the theories of $H_{\aleph _1}$ or $H_{\aleph _2}$ they entail.

In order to better argue for this form of justification of large cardinals and forcing axioms, let us briefly revise what we mean by intrinsic and extrinsic reasons.

• • A form of justification is considered intrinsic when it is based on intrinsic features of the concept of set or, derivatively (given the foundational role of set theory), on notions that are key to the whole edifice of mathematics and logic. In this sense, a new axiom is justified when it captures an essential aspect of the concept of set or when it formalizes notions that are fundamental for mathematics and logic. This form of justification is utterly conceptual, since it rests on the theoretical priority of the notion of set over its formalization. Consequently, an extension of $\mathsf {ZFC}$ is well justified, when it faithfully represents the correct concept of set (e.g., the iterative conception [Reference Boolos11] or the quasi-combinatorial one [Reference Bernays10]).

• • A form of justification is considered extrinsic when we are forced to accept an axiom by the abundance of its desirable consequences; even in the absence of intrinsic reasons. In this case an axiom is justified even if it does not seem prima facie to capture any relevant aspect of the concept of set. This second form of justification is clearly meant to overcome the limits of intrinsic reasons and to account for an experimental methodology in the context of the foundations of mathematics. Extrinsic justifications are pragmatic in nature, since they rely on the fruitfulness of an axiom in solving new problems, shortening proofs of already known results, and unifying substantial bodies of theory. The form of justification put forward by extrinsic reasons is akin to an inference to the best explanation within mathematics. Therefore, an extrinsically justified axiom is judged by its consequence and not by its meaning.

Intrinsic and extrinsic reasons have been extensively studied in the philosophy of set theory [Reference Maddy29, Reference Maddy30] and they have been widely applied, in recent debates, for the justifications of competing programs [Reference Arrigoni and Friedman2, Reference Koellner23]. These two forms of justification have also been criticized for their opacity in offering clear criteria of application and for the lack of demarcation between intrinsic and extrinsic reasons [Reference Barton, Ternullo and Venturi9]. Moreover, as it happened in the debate on the analytic-synthetic distinction, there is also no shortage of contributions that reject the problem of justification at its very base. Following a naturalistic account of mathematics, authors like Hamkins (or before him Cohen [Reference Cohen14]) propose to dismiss the problem of justification, together with the problem of independence, by defending the liberty of mathematicians to study different universes of set theory and by declaring the study of the variation of truth values among the models of $\mathsf {ZFC}$ to be all that mathematically matters for the study of independence [Reference Hamkins19].

Now, in which sense the nice model-theoretic properties of model companions are able to justify large cardinals and strong forcing axioms? Is the role of the latter set-theoretical principles in the individuation of model companions of set theory able to provide an intrinsic or an extrinsic justification for them?

A first complication that we face in addressing these questions is that the present approach does not only deal with axiomatic extensions of $\mathsf {ZFC}$ , but also with linguistic ones: the introduction of new signatures for set theory. This peculiar aspect is responsible for a justification of large cardinals and strong forcing axioms that contains elements of both intrinsic and extrinsic arguments. Indeed, on the one hand strong forcing axioms provide a maximization of the $\Pi _2$ -statements realized over $H_{\aleph _2}$ and, thus, have tremendously abundant consequences on third order arithmetic (and on this ground they can be extrinsically justified). On the other hand, because of the possibility to provide (generic) absoluteness results, large cardinal axioms determine the “simple” concepts that need to be included in the new signatures for set theory.Footnote ⁵⁷ Therefore, large cardinals and strong forcing axioms help us understanding what are the basic concepts on which third order arithmetic should be based (and for this reason they can be intrinsically justified).Footnote ⁵⁸

Another aspect of the present approach (that places the justification of new axioms somewhat outside the standard practice) is the focus on models, instead of sets (like intrinsic reasons), or problems (like extrinsic ones). As a matter of fact, large cardinals and strong forcing axioms play an important role in the construction of model companions of set theory, by detecting the natural signatures for which one has nice model-theoretic properties for (the models of) set theory. Consequently, the explicit link these axioms establish between the choice of a signature and model companionship results for set theory can also be seen as an indirect effect of these axioms in capturing certain natural features of the universe of sets. In this sense, the present approach makes evident the role that certain additional axioms of set theory have in clarifying the model theory of set theory, but also, and derivatively, the role that model-theoretic properties have in selecting what could be the natural theories for initial segments of the universe of sets. It is with respect to this double role (selecting simple concepts and providing natural theories) that our approach rooted in model companionship argues for the justifications of new axioms of set theory.

Because of the limits of the standard intrinsic–extrinsic dichotomy to capture the form of justification of this model-theoretic approach, we may call this new form of justification algebraic, since it rests on the model-theoretic properties of algebraically maximal theories. It is a form of justification that has both intrinsic and extrinsic elements and that heavily relies on the model-theoretic form of maximality that we described in the previous section.

In order to exemplify this algebraic form of justification, let us see how the present approach advances Gödel’s program in providing well-justified extensions of $\mathsf {ZFC}$ . This particular implementation of Gödel’s program is devised to provide new well-justified axioms of larger and larger initial segments of the universe V, concretely, by providing axioms able to make the theories of $H_{\kappa ^+}$ (as $\kappa $ ranges among the infinite cardinal) the model companions of set theory for the “correct” signature aimed at describing the elementary properties of $\mathcal {P}\left (\kappa \right )$ . In this way the theory of V is gradually completed by adding new principles whose justification rest on the nice model-theoretic properties they provide for the theories of the different $H_{\kappa ^+}$ . Let us see how this justification process can be implemented.

We start, of course, from $\mathsf {ZFC}$ and notice that this theory is algebraically maximal relative to $\in _A$ , which is the signature obtained by adding to $\in _{\Delta _0}$ predicates for the lightface definable subsets of in the Chang model .Footnote ⁵⁹ In this case its existentially complete models describe the theory of , and—because is a definable sub-model of V—we have that all condition of algebraic maximality are fulfilled. By [Reference Venturi and Viale38, theorem 5.2], this theory is the AMC of any extension of $\mathsf {ZFC}$ for $\in _A$ .