2.1 A term-calculus for representation of syntax

We shall work with various recursively enumerable set theories, in languages obtained by adding finitely many new non-logical symbols to the usual language of set theory on the signature $\{\in \}$ . We define a set theory to be any recursively enumerable system proving Mac Lane set theory (excluding Choice)Footnote ⁵ or proving Kripke–Platek set theory with Infinity,Footnote ⁶ in a language with finitely many non-logical symbols including a term-calculus for arithmetic and Gödel coding explained below. Our specific choice of set theories underlying this definition is somewhat arbitrary (even weaker theories may well suffice). Both of these theories are sufficient for constructing the structure of the natural numbers and implementing basic model theory; these are the important features for this paper.

Since we will be reasoning about syntactic objects, it is convenient to employ a Gödel coding of syntax. Let K be a language with finitely many non-logical symbols. In any set theory T (in language L) under consideration, we can define the arithmetic functions needed to formulate a natural Gödel coding in T of terms and formulas of K. Through the Gödel coding, the “grammatical structure” of K is coherently represented in T. The complicated details of this procedure are described in any rigorous account of Gödel’s incompleteness theorems. The gist is that for each syntactic object (symbol, term or formula) s of K, there is a definable number $\ulcorner s \urcorner $ in L (the Gödel code of s), which represents s in T, and there are operations definable in T corresponding to syntactic operations on such objects. The authors trust that the reader is familiar with this.

It is customary in set theory to informally introduce defined constant, relation and function symbols to the language, in order to make the presentation more readable. For example, one may use a function symbol $+$ , as if it belonged to the language and there was an axiom expressing that $+$ is addition on the finite von Neumann ordinals. In this paper we assume that a finite number of such symbols needed for arithmetic and Gödel coding are already present in the language of every set theory, and that the appropriate axioms regulating them are available in every set theory. Here follows a semi-formal account of some of the main principles of this expanded language L for a set theory T, also serving to specify the notation:

1. 1. We have a constant $\underline {0}$ and function symbols $S, +, \times $ for the successor, addition and multiplication operations in arithmetic. For each $n \in \mathbb {N}$ , $\underline {n}$ is shorthand for $S^n(\underline {0})$ .

2. 2. Each variable, constant, relation or function symbol s of K is represented in T by a numeral $\ulcorner s \urcorner $ in L.

3. 3. Recursively, each term $f(t_1, \dots , t_n)$ of K is represented by the term $\ulcorner f \urcorner (\ulcorner t_1 \urcorner , \dots , \ulcorner t_n \urcorner )$ of L. Formally, the term $\ulcorner f \urcorner (\ulcorner t_1 \urcorner , \dots , \ulcorner t_n \urcorner )$ is the result of applying a function symbol of L to the numerals $\ulcorner f \urcorner , \ulcorner t_1 \urcorner , \dots , \ulcorner t_n \urcorner $ . Moreover, $\ulcorner f(t_1, \dots , t_n) \urcorner $ denotes a numeral that T proves to equal $\ulcorner f \urcorner (\ulcorner t_1 \urcorner , \dots , \ulcorner t_n \urcorner )$ .

4. 4. Each atomic formula $R(t_1, \dots , t_n)$ of K is represented by the numeral $\ulcorner R(t_0, \dots , t_n) \urcorner $ of L. Analogous remarks apply as in the case of terms described above.

5. 5. The syntactic operations, standardly used to build up complex formulas from atomic formulas, are all available. For example:

   1. (a) L has a function symbol $\mathop {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\neg }}$ , such that for each $\phi $ in K, $\mathop {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\neg }} \ulcorner \phi \urcorner $ represents $\neg \phi $ . Moreover, T proves that $\ulcorner \neg \phi \urcorner =\mathop {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\neg }} \ulcorner \phi \urcorner $ .

   2. (b) L has a function symbol $\mathrel {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\wedge }}$ , such that for each $\phi $ and each $\psi $ in K, $\ulcorner \phi \urcorner \mathrel {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\wedge }} \ulcorner \psi \urcorner $ represents $\phi \wedge \psi $ . Moreover, T proves that $\ulcorner \phi \wedge \psi \urcorner =\ulcorner \phi \urcorner \mathrel {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\wedge }} \ulcorner \psi \urcorner $ .

   3. (c) L has a function symbol $\mathop {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\forall }}$ , such that for each variable v and each formula $\phi $ in K, $\mathop {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\forall }} \ulcorner v \urcorner \hspace {2pt} \ulcorner \phi \urcorner $ represents $\forall v \hspace {2pt} \phi $ . Moreover, T proves that $\ulcorner \forall v \hspace {2pt} \phi \urcorner =\mathop {\underset {\substack {\\[-9.5pt]\mbox {.}}}{\forall }} \ulcorner v \urcorner \hspace {2pt} \ulcorner \phi \urcorner $ .

   4. (d) For any $\phi $ in K, $\phi [t/x]$ denotes the formula obtained from $\phi $ by replacing each free occurrence of the variable x by the term t (if t has variables, then their bound occurrences in $\phi $ are renamed as necessary). L has a function symbol (written $-[-\underset {\substack {\\[-9.5pt]\mbox {.}}}{/}-]$ ) which represents this primitive recursive substitution operation. Moreover, T proves that $\ulcorner \phi (x)[y/x] \urcorner = \ulcorner \phi (x) \urcorner [\ulcorner y \urcorner \underset {\substack {\\[-9.5pt]\mbox {.}}}{/}\ulcorner x \urcorner ]$ . Somewhat less formally, if $\phi $ has been introduced as $\phi (x)$ , we may write $\phi (t)$ for the formula $\phi [t/x]$ .

In the context of a set theory T in a set-theoretic language L, $\Sigma ^0_n$ , $\Pi ^0_n$ and $\Delta ^0_n$ denote the usual arithmetic hierarchy as defined for L-formulas (all quantifiers are bounded to $\mathbb {N}$ ), up to equivalence in T. It is well-known that for any recursive system S, there is a $\Sigma ^0_1$ -formula $\Pr _{\underset {\substack {\\[-9.5pt]\mbox {.}}}{S}}$ , representing S-provability in T. We write $\mathrm {Con}_{\underset {\substack {\\[-9.5pt]\mbox {.}}}{S}}$ for the sentence $\neg \Pr _{\underset {\substack {\\[-9.5pt]\mbox {.}}}{S}}(\ulcorner \bot ) \urcorner $ , expressing that S is consistent. In both cases, the dot under S is sometimes omitted, when it can be inferred from the context.

As an example, consider this consequence of Gödel’s second incompleteness theorem:

$$ \begin{align*} \mathsf{ZF} \not\vdash \mathrm{Con}_{\underset{\substack{\\[-9.5pt]\mbox{.}}}{\mathsf{ZF}}}. \end{align*} $$

From the perspective of the meta-theory, “ $\mathsf {ZF}$ ” refers to a set of sentences (the object theory of $\mathsf {ZF}$ ) whereas “ $\underset {\substack {\\[-9.5pt]\mbox {.}}}{\mathsf {ZF}}$ ” refers to a formula representing the recursive set of Gödel codes of that set in the object theory $\mathsf {ZF}$ .

Suppose now that a set theory $T^{\prime }$ in language $L^{\prime }$ is represented in a set theory T in language L. Then $L^{\prime }$ is Gödel coded in T, as explained above. But $T^{\prime }$ , in turn, also Gödel codes languages; say that $T^{\prime }$ Gödel codes the language $L^{\prime \prime }$ . Note that the whole Gödel coding of $L^{\prime \prime }$ in $T^{\prime }$ is then carried along by the representation of $T^{\prime }$ in T. For example, if $\phi $ is a formula in $L^{\prime \prime }$ , then there is a term $\ulcorner \phi \urcorner $ in $L^{\prime }$ which represents $\phi $ in $T^{\prime }$ . If $\psi (x)$ is a formula of $L^{\prime }$ , we can then form the formula $\psi [\ulcorner \phi \urcorner / x]$ of $L^{\prime }$ . This formula, in turn, is represented in T by an L-term $\ulcorner \psi [\ulcorner \phi \urcorner / x] \urcorner $ . So if $\theta (y)$ is an L-formula, we can form the L-formula $\theta [\ulcorner \psi [\ulcorner \phi \urcorner / x] \urcorner / y]$ . Thus, Gödel codes may be nested, as a set theory represents a set theory, which in turn represents a language.

2.2 Miscellaneous logical preliminaries

$\mathcal {L}$ is the language with the symbol “ $\in $ ” along with a finite number of arithmetic and syntactic symbols as explained in Section 2.1. We assume that $\mathsf {ZF}$ is formulated as an $\mathcal {L}$ -theory, with the natural axioms for defining the arithmetic and syntactic symbols of $\mathcal {L}$ . $\mathcal {L}^+$ denotes any extension of $\mathcal {L}$ with a finite number of new symbols.

If L is a language and $S_1, \ldots , S_n$ are symbols, then $L_{S_1, \ldots , S_n}$ denotes the language obtained by augmenting L with $S_1, \ldots , S_n$ . The Separation schema applying to all formulas of a language L is denoted $\mathsf {Sep}(L)$ , and the Replacement schema applying to all formulas of a language L is denoted $\mathsf {Rep}(L)$ .

Any set theory suffices as meta-theory. Suppose that in the meta-theory we consider a definable set $A = \{x \mid \phi (x)\}$ , such as a theory. We may then refer to the corresponding set within an object-theory, for example as follows: Using the symbol A somewhat ambiguously, we write a statement of the form $\mathsf {ZF} \vdash \cdots \mathcal {M} \models \underset {\substack {\\[-9.5pt]\mbox {.}}}{A} \cdots $ for the more formally precise statement of the form $\mathsf {ZF} \vdash \cdots \exists X \hspace {2pt} \big ( \forall x (x \in X \leftrightarrow \phi (x)) \wedge \forall x \in X \hspace {2pt} (\mathcal {M} \models x) \big ) \cdots $ . The dot under A is occasionally omitted, when clear from the context. To illustrate, we might express a special case of Gödel’s completeness theorem within T as $T \vdash \big ( \mathrm {Con}_{{\mathsf {ZF}}} \leftrightarrow \exists \mathcal {M} (\mathcal {M} \models {\mathsf {ZF}}) \big )$ . We say that a theory $T_1$ bounds the consistency strength of (or has at least as high consistency strength as) a theory $T_0$ if the consistency of $T_1$ implies the consistency of $T_0$ .

An interpretation $\mathcal {I}$ from a language $L_0$ to a language $L_1$ is a function $\mathcal {I} : L_0 \rightarrow L_1$ which is generated (by structural recursion) from the non-logical symbols of $L_0$ . Moreover, we say that $\mathcal {I}$ interprets or validates the $L_0$ -system $T_0$ in the $L_1$ -system $T_1$ if for any $L_0$ -formula $\phi $ , $T_1 \vdash \mathcal {I}(\phi )$ whenever $T_0 \vdash \phi $ .

As default, we work with first-order languages and classical logic, but we will consider additional deductive rules (NEC and CONEC). $\phi \equiv \psi $ is the statement that $\phi $ and $\psi $ are identical formulas. If S and T are systems in languages both including L, then $S \equiv _L T$ is the statement that S and T have the same L-theorems. If S is a system involving deductive rules, and A is an axiom, then $S + A$ denotes the natural extension of S in which these deductive rules may be applied to proofs also involving A. For example, in $\mathsf {MS} + \exists x \hspace {2pt} \mathsf {Uni}(x)$ , we may use $\mathsf {NEC}$ to derive $\forall \mathcal {U} \in \mathsf {Uni}\hspace {2pt} \mathsf {Mod}(\mathcal {U}, \ulcorner \exists x \hspace {2pt} \mathsf {Uni}(x) \urcorner )$ .

It is sometimes notationally convenient to introduce classes of the form $A = \{ x \mid \phi (x) \}$ (the class of all sets x such that $\phi (x)$ ), where $\phi $ is an $\mathcal {L}^+$ -formula. Then $x \in A$ may be regarded as an alternative notation for $\phi (x)$ . Thus, we have no need to specify a formal class theory. For example $V = \{ x \mid \top \}$ is the class of all sets.

Formally, $\mathrm {Var}$ is the set of variables $\{{x}_1, {x}_2, \dots \}$ , indexed by the positive natural numbers. But we use other symbols informally (such as $x, y, p, f, \mathcal {U}, \ldots $ ) for variables as well. $\mathrm {VA}$ is the class of variable-assignments, $\{f \mid f : \mathrm {Var} \rightarrow V \}$ . If a is a set (or a structure), then $\mathrm {VA}^a$ is the set $\{f \mid f : \mathrm {Var} \rightarrow a\}$ of all variable-assignments to elements of (the underlying set of) a. If f is a variable-assignment and v is a variable, then $\mathrm {VA}_{f, v}$ is the set of all variable-assignments g, such that for all $u \in \mathrm {Var}$ , $u \neq v \rightarrow g(u) = f(u)$ . Suppose that we are working in a set theory T in a language L containing terms $t_1, \ldots t_n$ . Note that for $n < \omega $ , T proves from $v_1 \in \mathrm {Var}, \ldots , v_n \in \mathrm {Var}$ that there is a primitive recursive variable-assignment f satisfying $f(v_1) = t_1, \ldots , f(v_n) = t_n$ , and $\forall m \in \mathbb {N} \hspace {2pt} (m> n \rightarrow f(v_m) = \underline {0})$ . Such a variable assignment f is denoted $\langle v_1, \dots v_n \rangle \mapsto \langle t_1, \ldots t_n \rangle $ (or just $v_1 \mapsto t_1$ in the case $n = 1$ ).

We assume that model theory is set up so that any structure $\mathcal {M}$ uniquely determines its language, which we denote by $L(\mathcal {M})$ , and we take the symbol “ $\mathcal {M}$ ” to refer ambiguously to both the structure and its domain. Let $\mathcal {M}$ be an L-structure, $\phi $ a formula in L and $f \in \mathrm {VA}^{\mathcal {M}}$ . We use the arrow-notation $\vec {a}$ for finite tuples $\langle a_1, \ldots , a_n \rangle $ , and the shorthand $\vec {a} \in \mathcal {M}$ for that each component $a_i$ of $\vec {a}$ is an element of $\mathcal {M}$ . We write $\mathcal {M} \models (\phi , f)$ for the statement “ $\phi $ is true in $\mathcal {M}$ under the variable-assignment f,” as defined in the usual Tarskian semantics of first-order logic. If $\vec {a} \in \mathcal {M}$ and $\psi (\vec {x}) \in L$ , then we write $\mathcal {M} \models \psi (\vec {a})$ for $\mathcal {M} \models (\psi , \vec {x} \mapsto \vec {a})$ . We write $\mathcal {M} \models \phi $ for $\forall f \in \mathrm {VA}^{\mathcal {M}} \hspace {2pt} \mathcal {M} \models (\phi , f)$ . If K is a sublanguage of L, then $\mathcal {M}\hspace {-4pt}\restriction _K$ denotes the reduct of $\mathcal {M}$ to K. We write $\mathcal {M} \equiv _K \mathcal {N}$ for the statement that $\mathcal {M}$ satisfy the same K-sentences as $\mathcal {N}$ . We write $\mathcal {M} \cong _K \mathcal {N}$ for the statement that $\mathcal {M}\hspace {-4pt}\restriction _K$ is isomorphic to $\mathcal {N}\hspace {-4pt}\restriction _K$ . When the subscripts are dropped, they are assumed to be $L(\mathcal {M})$ .

We use abbreviations for certain variations of the quantifiers:

1. 1. $\exists x \in y \hspace {2pt} \phi $ stands for $\exists x \hspace {2pt} (x \in y \wedge \phi )$ .

2. 2. $\forall x \in y \hspace {2pt} \phi $ stands for $\forall x \hspace {2pt} (x \in y \rightarrow \phi )$ .

3. 3. $\exists ^! x \hspace {2pt} \phi $ stands for $\exists x \hspace {2pt} (\phi (x) \wedge \forall y \hspace {2pt} (\phi (y) \rightarrow x = y))$ .

If P is a predicate symbol, we may write $x \in P$ for $P(x)$ . Similarly, we write $\exists x \in P \hspace {2pt} \phi $ for $\exists x \hspace {2pt} (P(x) \wedge \phi )$ , and so on.

We will introduce primitive relation symbols “ $\mathsf {Sat}$ ,” “ $\mathsf {Uni}$ ” and “ $\mathsf {Mod}$ .” Informally, $\mathsf {Sat}(\phi , f)$ expresses that $\phi $ is satisfied under the variable assignment f; $\mathsf {Uni}(\mathcal {U})$ expresses that $\mathcal {U}$ is a universe; and $\mathsf {Mod}(\mathcal {U}, \phi , f)$ expresses that $\phi $ is satisfied in the universe $\mathcal {U}$ under the variable assignment f. Recall that $\mathcal {L}_{\mathsf {Sat}}$ denotes the language $\mathcal {L}$ augmented with the symbol $\mathsf {Sat}$ , while $\mathcal {L}_{\mathsf {Uni}, \mathsf {Mod}}$ denotes $\mathcal {L}$ augmented with $\mathsf {Mod}$ and $\mathsf {Uni}$ .

Again, we introduce some abbreviations:

1. 1. $\mathsf {Sat}(\phi )$ and $\mathsf {Tr}(\phi )$ stand for $\forall f \in \mathrm {VA} \hspace {2pt} \mathsf {Sat}(\phi , f)$ .

2. 2. $\mathsf {Mod}(\mathcal {U}, \phi )$ stands for $\forall f \in \mathrm {VA}^{\mathcal {U}} \hspace {2pt} \mathsf {Mod}(\mathcal {U}, \phi , f)$ .

3. 3. $\mathsf {Tr}^\Box (\phi )$ stands for $\forall \mathcal {U} \in \mathsf {Uni}\hspace {2pt} \forall f \in \mathrm {VA}^{\mathcal {U}} \hspace {2pt} \mathsf {Mod}(\mathcal {U}, \phi , f)$ .

4. 4. $\mathsf {Tr}^\Diamond (\phi )$ stands for $\neg \mathsf {Tr}^\Box (\dot \neg \phi )$ .

If X is a formula, term or definable object in the language L of the structure $\mathcal {M}$ , then $X^{\mathcal {M}}$ denotes its interpretation in $\mathcal {M}$ ; e.g., $\phi ^{\mathcal {M}} =_{\mathrm {df}} \{\vec {a} \in \mathcal {M} \mid \mathcal {M} \models \phi (\vec {a})\}$ .

Informally speaking, if $\mathcal {M}$ is a model of set theory, $\mathcal {N} \in \mathcal {M}$ , and $\mathcal {M} \models $ “ $\mathcal {N}$ is a model of set theory,” so that $\mathcal {N}$ is an internal model of $\mathcal {M}$ , then we may need to extract $\mathcal {N}$ into an external model that can be studied on a par with $\mathcal {M}$ . This is achieved by the following formal definition.

Definition 2.1. If $\mathcal {M}$ is a model of set theory and $a \in \mathcal {M}$ , then

$$ \begin{align*} a_{\mathcal{M}} =_{\mathrm{df}} \{x \in \mathcal{M} \mid \mathcal{M} \models x \in a\}. \end{align*} $$

This notation is generalized in the cases that a is considered as a relation or as a structure: If $\mathcal {M}$ is a model of set theory, $R \in \mathcal {M}$ and $\mathcal {M}$ satisfies that R is an $\underline {n}$ -ary relation (for a natural number n under consideration), then

$$ \begin{align*} R_{\mathcal{M}} =_{\mathrm{df}} \big\{ \langle x_1, \ldots, x_n \rangle \mid \hspace{2pt} & \exists p \in \mathcal{M} \hspace{2pt} [ \mathcal{M} \models p \in R \hspace{2pt} \wedge \\ & \bigwedge_{1 \leq i \leq n} \text{“}x_i \text{ is the }\underline{i}\text{:th component of }p”]\big\}. \end{align*} $$

If $\mathcal {M}$ is a model of set theory, $\mathcal {N}, N, R_1, \ldots , R_n$ are elements of $\mathcal {M}$ , and $\mathcal {M}$ satisfies that $\mathcal {N}$ is a structure with domain N and relations $R_1, \ldots , R_n$ (of arities $\underline {r_1}, \ldots , \underline {r_n}$ , respectively), then

$$ \begin{align*} \mathcal{N}_{\mathcal{M}} =_{\mathrm{df}} \langle N_{\mathcal{M}}, (R_1)_{\mathcal{M}}, \ldots, (R_n)_{\mathcal{M}} \rangle. \end{align*} $$

In either of the above cases $a_{\mathcal {M}}$ is called the $\mathcal {M}$ -externalization of a.

For example, $\omega ^{\mathcal {M}}$ denotes the element of $\mathcal {M}$ such that $\mathcal {M} \models \phi (\omega )$ , where $\phi $ defines $\omega $ . On the other hand, $(\omega ^{\mathcal {M}})_{\mathcal {M}}$ denotes the subset $\{a \in \mathcal {M} \mid \mathcal {M} \models a \in \omega \}$ of $\mathcal {M}$ , consisting of all a such that $\mathcal {M} \models a \in \omega $ .

2.3 Recursive saturation

A type $p(\vec {x})$ , over a theory T in a language L, is a set of L-formulas such that $T \cup p$ is consistent when the variables $\vec {x}$ are considered as fresh constant symbols. If $\mathcal {M} \models T$ , then $p(\vec {x})$ is realized in $\mathcal {M}$ if there is $\vec {a} \in \mathcal {M}$ , such that for all $\phi (\vec {x}) \in p$ , we have $\mathcal {M} \models \phi (\vec {a})$ . A type $p(\vec {x}, \vec {b})$ , over $\mathcal {M}$ with parameters $\vec {b} \in \mathcal {M}$ , is a type over $\mathrm {Th}(\mathcal {M}, \vec {b})$ (the theory of $\mathcal {M}$ with parameters $\vec {b}$ ). Such a type $p(\vec {x}, \vec {b})$ is recursive if it is a recursive set (under some fixed Gödel coding of the formulas as natural numbers). A structure $\mathcal {M}$ is recursively saturated if it realizes every recursive type over $\mathcal {M}$ . A $\text {crsm}$ is a countable recursively saturated model.

Let $\mathcal {M}$ be a model of a set theory T. The interpretations of the numerals in $\mathcal {M}$ are called the standard natural numbers of $\mathcal {M}$ . For each $n < \omega $ , let us make the identification $n = \underline {n}^{\mathcal {M}}$ . We say that $\mathcal {M}$ is $\omega $ -non-standard if there is $c \in (\omega ^{\mathcal {M}})_{\mathcal {M}} \setminus \omega $ . Such a c is said to be a non-standard number of $\mathcal {M}$ .

Suppose that $\mathcal {M}$ is $\omega $ -non-standard. We say that a subset A of $\omega $ is coded in $\mathcal {M}$ , if there is (a code) $a \in (\omega ^{\mathcal {M}})_{\mathcal {M}}$ , such that $A = \{n \in \mathbb {N} \mid \mathcal {M} \models \underline {n} < a\}$ . We define the standard system of $\mathcal {M}$ as

$$ \begin{align*} \mathrm{SSy}(\mathcal{M}) =_{\mathrm{df}} \{A \subseteq \omega \mid \text{“}A \text{ is coded in } \mathcal{M}\text{”}\}. \end{align*} $$

The following result is due to [Reference Wilmers17], employing Friedman’s back-and-forth technique [Reference Friedman, Mathias and Rogers4]:

We define the class

$$ \begin{align*} \mathbf{crsm} =_{\mathrm{df}} \{\mathcal{M} \mid \mathcal{M}\hspace{-4pt}\restriction_{\mathcal{L}} \models \mathsf{ZF} \wedge “\mathcal{M} \text{ is a } \text{crsm}''\}. \end{align*} $$

2.4 Systems of satisfaction over $\mathsf {ZF}$

Before embarking on developing a framework for a notion of truth-in-a-universe relevant to the set-theoretic multiverse, we shall go through some related systems of truth over $\mathsf {ZF}$ . An intuitive philosophical perspective is that systems of truth capture various absolute notions of truth, as motivated by the universe view of set theory, while our framework for truth-in-a-universe captures various relative notions of truth, as motivated by the multiverse view of set theory. From a mathematical perspective, it is interesting to relate these two approaches. Moreover, since we will generalize techniques that have been developed for studying systems of truth, these provide a relevant context for viewing our results.

Right at the start of the endeavour to axiomatize truth, one faces the choice between introducing (to the base language of set theory) a unary truth-predicate $\mathsf {Tr}(\sigma )$ , applying to sentences $\sigma $ , or a binary satisfaction-relation $\mathsf {Sat}(\phi , f)$ , applying to formulas $\phi $ and variable-assignments $f : \mathrm {Var} \rightarrow V$ . In the former option, $\sigma $ needs to range over a class-sized language where there is a constant-symbol $c_x$ corresponding to each $x \in V$ . The authors have chosen the latter option. As a general heuristic, one is justified to expect any theory of satisfaction to be interpretable in a corresponding theory of truth; the idea being to interpret $\mathsf {Sat}(\phi (x), f)$ by $\mathsf {Tr}(\phi [c_{f(x)}/x])$ . Fujimoto has made a comprehensive study of theories of truth over set theory, following the former option [Reference Fujimoto6].

System 2.5 ( $\mathsf {CT}$ )

Let S be a set theory in $\mathcal {L}^+$ . The system $\mathsf {CT}(S)\hspace {-4pt}\restriction $ (for Compositional Truth) consists of these axioms in the language $\mathcal {L}^+_{\mathsf {Sat}}$ :

$$ \begin{align*} \begin{array}{ll} \mathsf{Base} & S, \\ \mathsf{CT}_= & \forall y_0, y_1 \hspace{2pt} \big( \mathsf{Sat}(\ulcorner x_0 = x_1 \urcorner, \langle \ulcorner x_0 \urcorner, \ulcorner x_1 \urcorner \rangle \mapsto \langle y_0, y_1 \rangle) \leftrightarrow y_0 = y_1 \big), \\ \mathsf{CT}_\in & \forall y_0, y_1 \hspace{2pt} \big( \mathsf{Sat}(\ulcorner x_0 \in x_1 \urcorner, \langle \ulcorner x_0 \urcorner, \ulcorner x_1 \urcorner \rangle \mapsto \langle y_0, y_1 \rangle) \leftrightarrow y_0 \in y_1 \big), \\ \mathsf{CT}_\neg & \forall \phi \in \mathcal{L}^+_{\mathsf{Sat}} \hspace{2pt} \forall f \in \mathrm{VA} \hspace{2pt} (\mathsf{Sat}(\underset{\substack{\\[-9.5pt]\mbox{.}}}{\neg} \phi, f) \leftrightarrow \neg \mathsf{Sat}(\phi, f)), \\ \mathsf{CT}_\wedge & \forall \phi, \psi \in \mathcal{L}^+_{\mathsf{Sat}} \hspace{2pt} \forall f \in \mathrm{VA} \hspace{2pt} (\mathsf{Sat}(\phi \mathrel{\underset{\substack{\\[-9.5pt]\mbox{.}}}{\wedge}} \psi, f) \leftrightarrow \mathsf{Sat}(\phi, f) \wedge \mathsf{Sat}(\psi, f)), \\ \mathsf{CT}_\forall & \forall \phi \in \mathcal{L}^+_{\mathsf{Sat}} \hspace{2pt} \forall f \in \mathrm{VA} \hspace{2pt} (\mathsf{Sat}(\underset{\substack{\\[-9.5pt]\mbox{.}}}{\forall} u \hspace{2pt} \phi, f) \leftrightarrow \forall g \in \mathrm{VA}_{f, u} \hspace{2pt} \mathsf{Sat}(\phi, g)). \\ \end{array} \end{align*} $$

We write $\mathsf {CT}\hspace {-4pt}\restriction $ for $\mathsf {CT}(\mathsf {ZF})\hspace {-4pt}\restriction $ . The axioms of the form $\mathsf {CT}_-$ are called compositional axioms. By basic logic, $\mathsf {CT}\hspace {-4pt}\restriction $ also proves the axioms $\mathsf {CT}_\vee $ , $\mathsf {CT}^\rightarrow $ and $\mathsf {CT}_\exists $ (analogously defined). We use phrases such as “ $\mathsf {Sat}$ is $\vee $ -compositional” to express that we have $\mathsf {CT}_\vee $ , for example.

$\mathsf {CT}$ is $\mathsf {CT}\hspace {-4pt}\restriction + \hspace {2pt} \mathsf {Sep}(\mathcal {L}_{\mathsf {Sat}}) + \mathsf {Rep}(\mathcal {L}_{\mathsf {Sat}})$ .

A routine induction argument in the meta-theory shows this proposition:

The theory of satisfaction $\mathsf {CT}\hspace {-4pt}\restriction + \hspace {2pt} \mathsf {Sep}(\mathcal {L}_{\mathsf {Sat}})$ corresponds to the theory of truth $\mathsf {TC}\hspace {-4pt}\restriction + \hspace {2pt} \mathsf {Sep}^+$ in [Reference Fujimoto6, sec. Reference Friedman, Mathias and Rogers4]. It is straightforward to interpret the former in the latter. Using this, it follows from Theorem 20 in [Reference Fujimoto6, sec. Reference Friedman, Mathias and Rogers4.Reference Broberg1] that $\mathsf {CT}\hspace {-4pt}\restriction + \hspace {2pt} \mathsf {Sep}(\mathcal {L}_{\mathsf {Sat}})$ is conservative over $\mathsf {ZF}$ . In contrast, in $\mathsf {CT}$ we have access to the Reflection theorem for $\mathcal {L}_{\mathsf {Sat}}$ -formulas, enabling us to prove that there is a $V_\alpha $ modeling $\mathsf {ZF}$ . See [Reference Fujimoto6, sec. Reference Friedman, Mathias and Rogers4.Reference Broberg1] for more details and refinements.

System 2.8 ( $\mathsf {GR}^\omega $ )

Let S be a set theory in $\mathcal {L}^+$ . Here we present the systems of iterated Global Reflection over S, denoted $\mathsf {GR}^\alpha (S)$ , for ordinals $\alpha \leq \omega $ . For any set theory T in the language $\mathcal {L}_{\mathsf {Sat}}$ , the axiom of Global Reflection over T is

$$ \begin{align*} \begin{array}{ll} \mathsf{GR}_{\underset{\substack{\\[-9.5pt]\mbox{.}}}{T}} & \forall \phi \in \underset{\substack{\\[-9.5pt]\mbox{.}}}{\mathcal{L}}^+_{\mathsf{Sat}} \hspace{2pt} (\mathrm{Pr}_{\underset{\substack{\\[-9.5pt]\mbox{.}}}{T}}(\phi) \rightarrow \mathsf{Tr}(\phi)). \end{array} \end{align*} $$

(The dot under T is sometimes omitted, when it can be inferred from the context.)

Recursively, for each $\alpha \leq \omega $ , we define the system $\mathsf {GR}^\alpha (S)$ :

$$ \begin{align*} \mathsf{GR}^0(S) &=_{\mathrm{df}} \mathsf{CT}(S) \hspace{-4pt}\restriction + \hspace{2pt} \mathsf{Sep}(\mathcal{L}^+_{\mathsf{Sat}}), \\ \mathsf{GR}^{\alpha+1}(S) &=_{\mathrm{df}} \mathsf{GR}^\alpha(S) + \mathsf{GR}_{\mathsf{GR}^{\alpha}}, \\ \mathsf{GR}^\omega(S) &=_{\mathrm{df}} \bigcup_{n < \omega} \mathsf{GR}^n(S). \end{align*} $$

We write $\mathsf {GR}^\alpha $ for $\mathsf {GR}^\alpha (\mathsf {ZF})$ .

System 2.9 ( $\mathsf {FS}$ )

The systems $\mathsf {FS}\hspace {-3pt}\restriction $ and $\mathsf {FS}$ (for Friedman–Sheard) are obtained by adding these rules of proof to $\mathsf {CT}\hspace {-3pt}\restriction $ and $\mathsf {CT}$ , respectively:

$$ \begin{align*} \begin{array}{ll} \mathsf{NEC} & \text{For each } \phi \in \mathrm{Sent}(\mathcal{L}_{\mathsf{Sat}}): \text{ If } \mathsf{FS} \vdash \phi, \text{ then } \mathsf{FS} \vdash \mathsf{Tr}(\ulcorner \phi \urcorner). \\ \mathsf{CONEC} & \text{For each } \phi \in \mathrm{Sent}(\mathcal{L}_{\mathsf{Sat}}): \text{ If } \mathsf{FS} \vdash \mathsf{Tr}(\ulcorner \phi \urcorner), \text{ then } \mathsf{FS} \vdash \phi. \\ \end{array} \end{align*} $$

Recall that $\mathsf {Tr}(\theta )$ is defined as $\forall f \in \mathrm {VA} \hspace {2pt} \mathsf {Sat}(\theta , f)$ .

Given a set-theoretic system S in language L, the following rule will be considered:

$$ \begin{align*} \begin{array}{ll} \textsf{Reflection rule} & \text{For each } \phi \text{ in } L: \text{ If } S \vdash \Pr_S(\ulcorner \phi \urcorner), \text{ then } S \vdash \phi. \end{array} \end{align*} $$

Definition 2.10. A set-theoretic system S in language L is $\omega $ -inconsistent if there is an L-formula

such that:

We say that

witnesses the $\omega $ -inconsistency of S.Footnote ⁷

Let c be a fresh constant symbol. Note that the schema $\{\underline {m} < c < \omega \mid m \in \mathbb {N}\}$ , expressing that there is a non-standard natural number, yields $\omega $ -inconsistency when added to a set-theoretic system (proof: take $x = c$ as ).

So if we were to naturally extend the definition of $\mathsf {GR}^\alpha $ to all ordinals $\alpha $ , then we would get that $\mathsf {GR}^\alpha $ is inconsistent for all $\alpha> \omega $ .

Later on we will introduce a multiverse axiom, called $\textsf {Self-Perception}$ , to the effect that the universe of the background theory is isomorphic (over $\mathcal {L}$ ) to one of its internal universes; this axiom is motivated by the idea that the universe of the background theory should be available in its multiverse. The following lemmas establish technical results needed to validate that axiom.

Lemma 2.13. Let $\mathcal {U}$ be a $\text {crsm}$ of $\mathsf {GR}_0$ and let $\mathcal {V} \in \mathcal {U}$ , such that $\mathcal {U}$ satisfies

$$ \begin{align*} \mathcal{V} \models \{\sigma \in \mathrm{Sent}(\mathcal{L}_{\mathsf{Sat}}) \mid \mathsf{Tr}(\sigma)\}. \end{align*} $$

Then $\mathcal {U} \cong _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ .

Recall that $\mathcal {V}_{\mathcal {U}}$ is the $\mathcal {U}$ -externalization of $\mathcal {V}$ , see Definition 2.1.

Proof. We shall establish $\mathcal {U} \cong _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ by invoking Theorem 2.4. Thus we need to show that $\mathcal {V}_{\mathcal {U}}$ is a $\text {crsm}$ , that $\mathrm {SSy}(\mathcal {U}) = \mathrm {SSy}(\mathcal {V}_{\mathcal {U}})$ and that $\mathcal {U} \equiv _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ .

Note that $\mathcal {U}$ is $\omega $ -non-standard, by $\mathcal {U} \in \mathbf {crsm}$ and Proposition 2.3. That $\mathcal {V}_{\mathcal {U}}$ is a $\text {crsm}$ now follows from Lemma 2.2 in [Reference Gitman and Hamkins7].Footnote ⁸

$(\omega ^{\mathcal {U}})_{\mathcal {U}}$ is mapped initially into $(\omega ^{(\mathcal {V}_{\mathcal {U}})})_{(\mathcal {V}_{\mathcal {U}})}$ by an embedding j (for each $x \in (\omega ^{\mathcal {U}})_{\mathcal {U}}$ , $j(x)$ is defined as the unique $y \in \mathcal {U}$ such that $\mathcal {U} \models y = \underline {x}^{\mathcal {V}}$ ). Therefore, we obtain $\mathrm {SSy}(\mathcal {U}) = \mathrm {SSy}(\mathcal {V}_{\mathcal {U}})$ as follows: Let $A \in \mathrm {SSy}(\mathcal {U})$ , coded by $a \in (\omega ^{\mathcal {U}})_{\mathcal {U}}$ . Since j is an embedding, $j(a)$ is a code for A in $(\omega ^{(\mathcal {V}_{\mathcal {U}})})_{(\mathcal {V}_{\mathcal {U}})}$ . Conversely, let $B \in \mathrm {SSy}(\mathcal {V}_{\mathcal {U}})$ , coded by $b \in (\omega ^{(\mathcal {V}_{\mathcal {U}})})_{(\mathcal {V}_{\mathcal {U}})}$ . Since j is initial and $\mathcal {U}$ is $\omega $ -non-standard, there is a non-standard $c \in (\omega ^{\mathcal {U}})_{\mathcal {U}}$ , such that $j(c) \leq ^{\mathcal {V}_{\mathcal {U}}} b$ . So since j is an embedding, c is a code for B in $(\omega ^{\mathcal {U}})_{\mathcal {U}}$ .

To see $\mathcal {U} \equiv _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ , let $\phi $ be a sentence of $\mathcal {L}$ . By absoluteness of $\models $ for standard formulas,

$$ \begin{align*} \mathcal{V}_{\mathcal{U}} \models \phi \iff \mathcal{U} \models \text{“} \mathcal{V} \models (\ulcorner \phi \urcorner). \text{”} \end{align*} $$

Since $\mathsf {Sat}$ satisfies the Tarski-biconditionals for $\mathcal {L}$ , we have

$$ \begin{align*}\mathcal{U} \models \phi \iff \mathcal{U} \models \mathsf{Tr}(\ulcorner \phi \urcorner).\end{align*} $$

Moreover, by $\mathsf {CT}_\neg $ and the condition of the lemma,

$$ \begin{align*}\mathcal{U} \models \mathsf{Tr}(\ulcorner \phi \urcorner) \iff \mathcal{U} \models \text{“} \mathcal{V} \models (\ulcorner \phi \urcorner). \text{”}\end{align*} $$

By combining these we obtain $\mathcal {U} \equiv _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ , as desired.□

Lemma 2.14. Let $k < \alpha \leq \omega $ , and let $\mathcal {U} \models \mathsf {GR}^\alpha $ . Then there is $\mathcal {V} \in \mathcal {U}$ , such that $\mathcal {U}$ satisfies

$$ \begin{align*}\mathcal{V} \in \mathbf{crsm} \wedge \mathcal{V} \models \{\sigma \in \mathrm{Sent}(\mathcal{L}_{\mathsf{Sat}}) \mid \mathsf{Tr}(\sigma)\}.\end{align*} $$

In particular, $\mathcal {U}$ satisfies that $\mathcal {V} \models \mathsf {GR}^k$ . Moreover, if $\mathcal {U} \in \mathbf {crsm}$ , then $\mathcal {U} \cong _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ .

Axiom 2.15. Let $\iota $ be a function symbol and $\mathsf {self}$ be a constant symbol. $\mathrm {Iso}(x)$ denotes an $\mathcal {L}^+_\iota $ -formula expressing that x is an $\mathcal {L}^+$ -structure and that $\iota $ is an $\in $ -isomorphism from the universe V onto x. We shall study this axiom in $\mathcal {L}^+_{\iota , \mathsf {self}}$ :

$$ \begin{align*} \mathrm{Iso}(\mathsf{self}). \end{align*} $$

(This formulation is chosen over $\exists x \hspace {2pt} \mathrm {Iso}(x)$ , as it is convenient to have a reference to a witness.)

By the $\in $ -isomorphism property, and the absoluteness of $\models $ for standard formulas, we have:

System 2.17. Let S be a set theory in $\mathcal {L}^+$ . By recursion, for each $\alpha \leq \omega $ , we define the system $\mathsf {SP}^\alpha (S)$ (standing for Self-Perception):

$$ \begin{align*} \mathsf{SP}^0(S) &=_{\mathrm{df}} \mathsf{GR}^0(S) = \mathsf{CT}(S) \hspace{-4pt}\restriction + \hspace{2pt} \mathsf{Sep}(\mathcal{L}^+_{\mathsf{Sat}}), \\ \mathsf{SP}^{\alpha+1}(S) &=_{\mathrm{df}} \mathsf{GR}^{\alpha+1}(S) + \mathsf{self} \in \mathbf{crsm} + \mathrm{Iso}(\mathsf{self}) + \mathsf{self} \models \mathbf{Tr} \cup \underset{\substack{\\[-9.5pt]\mbox{.}}}{\mathsf{SP}}^{\underline{\alpha}}(\underset{\substack{\\[-9.5pt]\mbox{.}}}{S}), \\ \mathsf{SP}^\omega(S) &=_{\mathrm{df}} \bigcup_{n < \omega} \mathsf{SP}^n(S), \end{align*} $$

where $\mathbf {Tr} =_{\mathrm {df}} \{\sigma \in \mathrm {Sent}(\mathcal {L}_{\mathsf {Sat}}) \mid \mathsf {Tr}(\sigma )\}$ . We write $\mathsf {SP}^\alpha $ for $\mathsf {SP}^\alpha (\mathsf {ZF})$ .

Proof. We start by showing the case $\alpha < \omega $ by induction. The base case $\alpha = 0$ is trivial. The induction hypothesis is that if $\mathcal {U} \models \mathsf {GR}^{\alpha }$ , then $\mathcal {U}$ expands to a model of $\mathsf {SP}^\alpha $ ; and if $\mathcal {U}$ is definable, then the expansion is definable. Let $\mathcal {U} \models \mathsf {GR}^{\alpha + 1}$ . Applying Lemma 2.14, we find a $\text {crsm} \mathcal {V}$ in $\mathcal {U}$ , such that $\mathcal {U} \cong _{\mathcal {L}} \mathcal {V}_{\mathcal {U}}$ (as witnessed by an $\mathcal {L}$ -isomorphism $i : \mathcal {U} \rightarrow \mathcal {V}_{\mathcal {U}}$ ) and $\mathcal {U}$ satisfies that $\mathcal {V} \models \mathbf {Tr} \cup \underset {\substack {\\[-9.5pt]\mbox {.}}}{\mathsf {GR}}^{\underline {\alpha }}$ . So by the induction hypothesis applied in $\mathcal {U}$ , $\mathcal {U}$ satisfies that $\mathcal {V}$ expands to a model $\mathcal {W}$ of $\underset {\substack {\\[-9.5pt]\mbox {.}}}{\mathsf {SP}}^{\underline {\alpha }}$ . Let $\mathcal {U}^*$ be the model obtained from $\mathcal {U}$ by interpreting $\mathsf {self}$ by $\mathcal {W}$ and interpreting $\iota $ by i. It is immediate from the construction that $\mathcal {U}^* \models \mathsf {SP}^{\alpha + 1}$ .

Now to the case that $\alpha = \omega $ : Assume that $\mathcal {U} \models \mathsf {GR}^\omega $ . Working in $\mathcal {U}$ , let $\mathbf {Tr} = \{\phi \in \mathrm {Sent}(\mathcal {L}_{\mathsf {Sat}}) \mid \mathsf {Tr}(\phi )\}$ . By the above,

$$ \begin{align*} \{x \in \mathbf{crsm} \wedge x \models \mathbf{Tr} \cup \underset{\substack{\\[-9.5pt]\mbox{.}}}{\mathsf{SP}}^{\underline{n}} \mid n < \omega\} \end{align*} $$

is a recursive type over $\mathcal {U}$ . So since $\mathcal {U}$ is recursively saturated, it is realized by some $\mathcal {W}$ in $\mathcal {U}$ . It now follows from Lemma 2.13 that $\mathcal {U} \cong _{\mathcal {L}} \mathcal {W}_{\mathcal {U}}$ . So just as in the former case, $\mathcal {U}$ can be expanded to a model $\mathcal {U}^*$ of $\mathsf {SP}^\omega $ .

Assume now that $\mathcal {U}$ is definable. $\mathcal {W}$ can then be defined as the least element of a definable enumeration of $\mathcal {U}$ that satisfies the appropriate conditions. An isomorphism witnessing $\mathcal {U} \cong _{\mathcal {L}} \mathcal {W}_{\mathcal {U}}$ can also be defined: As seen from a close look at the proof of Theorem 2.4, the isomorphism is constructed by recursion on enumerations of $\mathcal {U}$ and $\mathcal {W}_{\mathcal {U}}$ , both of which can be chosen definable since $\mathcal {U}$ and $\mathcal {W}_{\mathcal {U}}$ are definable and countable.□