2.1 The difference between operations and functions

In standard textbooks on first-order theory, it is common to refer to the symbols in a signature of a first-order language as ‘relation symbols’ and ‘function symbols’. We cannot use this terminology here, since we reserve the words ‘relation’ and ‘function’ to their official meaning in set theory, that is, to sets of pairs satisfying certain conditions. Instead, we use the name ‘predicate’ for any “relation” that is not a set (like the predicates $\in $ or $=$ ), and we use the name ‘operation’ for any “function” that is not a set (like the operation of union on sets or the operation of addition on ordinals).Footnote ¹² Accordingly, the symbols of a first-order signature are divided in this paper into predicate symbols, operation symbols, and constants. The latter may actually be viewed as operations with arity 0, except that they should always be interpreted as sets.

2.2 Notations

We assume that every first-order language for a set theory has infinitely many variables for sets, officially taken here to be $v_0,v_1,\ldots $ . We use letters (small or capital) from the end of the Latin alphabet to vary over these variables, and letters from the beginning of that alphabet as general variables for sets in the metalanguage. As usual, $i,j,k,l,m,n$ are used as special variables for natural numbers (in both the metalanguage and the formal language). t and s serve as variables (in the metalanguage) for terms, and $\varphi ,\psi ,\theta $ (and sometimes also A and B) as variables for formulas. In all cases the variables may be decorated with subscripts or superscripts. We denote by $Fv(\varphi )$ ( $Fv(t)$ ) the set of free variables of $\varphi $ (of t). When we denote a formula by $\varphi (x_1,\ldots ,x_n)$ it means that $\{x_1,\ldots ,x_n\}\subseteq Fv(\varphi )$ . On the other hand, when we write $\varphi (\vec {y},x)$ it means that $\vec {y}=\langle y_1,\ldots ,y_n \rangle $ for some n (whose identity may be obtained from the context or it does not matter); the variables $x,y_1,\ldots ,y_n$ are all distinct from each other; and $Fv(\varphi )=\{x,y_1,\ldots ,y_n\}$ .

The substitution of a term t for a free variable y in a formula $\varphi $ (a term s) is denoted by $\varphi \{t/y\}$ ( $s\{t/y\}$ ). However, when we denote a formula by $\varphi (y)$ ( $\varphi (x,y)$ , $\varphi (\vec {x},y)$ ) we might simply write $\varphi (t)$ ( $\varphi (x,t)$ , $\varphi (\vec {x},t)$ ) instead.

Given a first-order signature $\sigma $ , we take a structure for $\sigma $ to be a pair $\mathcal {M}=\langle \mathcal {D},I \rangle $ , where $\mathcal {D}\neq \emptyset $ is the domain of $\mathcal {M}$ and I is its interpretation function. If r is one of the symbols in $\sigma $ we shall usually write $r^I$ instead of $I(r)$ . If $\nu $ is an assignment of elements of $\mathcal {D}$ to variables of the language, $x_1,\ldots ,x_n$ are n distinct variables, and $\vec {a}\in \mathcal {D}^n$ , we denote by $\nu \{\vec {x}:=\vec {a}\}$ the assignment which is obtained from $\nu $ by assigning $a_i$ to $x_i$ ( $i=1,\ldots ,n$ ). We denote by $\nu _{\mathcal {M}}[t]$ the element of $\mathcal {D}$ that $\nu $ assigns according to I to the term t of $\sigma $ . Similarly, if f is an operation symbol of $\sigma $ then we use in the metalanguage square brackets to denote applications of $f^I$ to arguments in $\mathcal {D}$ . Thus, if f is n-ary, and $\nu $ is an assignment in $\mathcal {D}$ , then $\nu _{\mathcal {M}}[f(t_1,\ldots ,t_n)]=f^I[\nu _{\mathcal {M}}[t_1],\ldots ,\nu _{\mathcal {M}}[t_n]]$ . We write $\mathcal {M},\nu \models \varphi $ in case $\nu $ satisfies in $\mathcal {M}$ the formula $\varphi $ of $\sigma $ . If $Fv(\varphi )=\{x_1,\ldots ,x_n\}$ , and $\nu [x_i]=a_i$ ( $i=1,\ldots ,n$ ) then we might write instead $\mathcal {M}\models \varphi (a_1,\ldots ,a_n)$ .

Finally, when we refer in the metalanguage to the collection of things that satisfy a certain condition C we shall denote it by $[x\mid C(x)]$ , reserving the notation $\{x\!\mid \!\varphi (x)\}$ for being used in our formal system. Moreover, in case there is a danger of confusion, we shall use ‘:’ in the metalanguage instead of ‘ $\in $ ’. (Recall that the latter is a basic symbol of the language of our system.)

4.1 A description of the system

The language of the system $\mathbf {PW}$ is an one-sorted first-order language with equality. As in Section 2, we take its variables to be $v_0,v_1,\ldots $ , and use letters (both small and capital) from the end of the Latin alphabet to vary over them. All other components of the language (predicate and operation symbols, terms, formulas, the invariance relation $\succ $ and the $\Sigma $ -formulas) are simultaneously generated as described in Sections 4.1.1–4.1.5. The axioms and rules of $\mathbf {PW}$ are presented in Sections 4.1.6 and 4.1.7, respectively. (Note that in the formulation of the last rule [Unif] there is a use of the formula $Fun$ , the operation $Dom$ , and the term $f(x)$ . They are all introduced in Section 5.1 without using [Unif]. $Fun(f)$ says that f is a function. $Dom(f)$ and $f(x)$ have their usual meanings.)

4.2 $\mathbf {PW}$ and $\mathbf {ZF}-(P)$

In this section we prove that $\mathbf {PW}$ is a subsystem of (an extension by definitions of) $\mathbf {ZF}-(P)$ . It follows that every theorem of $\mathbf {PW}$ is acceptable to the ordinary mathematicians.

Theorem 4.12. Let $\mathcal {M}$ be a transitive model of $\mathbf {ZF}-(P)$ , and let $\theta $ be a formula of $\sigma _{e}$ . Suppose that $\theta \succ \{x_1,\ldots ,x_n\}$ where $n>0$ . Then the following collection is an element of $\mathcal {M}$ for every assignment $\nu $ in $\mathcal {M}$ :

$$ \begin{align*}S_{\theta,\vec{x}}^{\nu}= [\vec{a}\in \mathcal{M}^n\mid \mathcal{M}^e,\nu\{\vec{x}=\vec{a}\}\models\theta].\end{align*} $$

4.3 Predicative justification of $\mathbf {PW}$

From the non-logical axioms and rules of $\mathbf {PW}$ , [Ext] and [PrI] need no justification, while [Inf] was justified in Section 3. It remains to justify the other non-logical axioms and rules of $\mathbf {PW}$ .

4.3.1 [Comp] and [OpI]

Theorem 4.12 alone is not sufficient for justifying [Comp]. In order to really justify this axiom from a predicative point of view, we should show that the identity of the sets it defines is invariant. Similarly, in order to justify [OpI] we should show that the operation defined by it is invariant. The first step towards these goals is to provide a precise notion of invariance which is adequate for the set-theoretical context.

Obviously, talking about invariance of definitions of sets and operations, when we expand one sphere of operation $\mathcal {M}_1$ to a bigger one $\mathcal {M}_2$ , can make sense only if the identities of the elements of $\mathcal {M}_1$ are preserved in $\mathcal {M}_2$ . Since in the context of set theory we take the identity of a set to be fully determined by the identity of its elements, this means that the same objects should belong to an element a of $\mathcal {M}_1$ in both $\mathcal {M}_1$ and $\mathcal {M}_2$ . Accordingly, we define the following:

Definition 4.18. Let $\mathcal {M}_1=\langle \mathcal {D}_1,I_1 \rangle $ and $\mathcal {M}_2=\langle \mathcal {D}_2,I_2 \rangle $ be structures for signatures that contain $\in $ , and let both be models of [Ext]. $\mathcal {M}_2$ is an $\in $ -extension of $\mathcal {M}_1$ if $\mathcal {D}_1\subseteq \mathcal {D}_2$ , and the following holds for every element a of  $\mathcal {D}_1$ :

$$ \begin{align*}[x:\mathcal{D}_1\mid x\in^{I_1}a]=[x:\mathcal{D}_2\mid x\in^{I_2}a].\end{align*} $$

Note 4.19. From a platonistic point of view, if $\in ^{I_1}$ and $\in ^{I_2}$ are both the ‘real’ $\in $ of $\mathsf {V}$ , then $\mathcal {M}_2$ is an $\in $ -extension of $\mathcal {M}_1$ iff $\mathcal {D}_1$ is a transitive subset of $\mathcal {D}_2$ .

Definition 4.20. For legal e, $\langle \mathcal {M}_1,\mathcal {M}_2 \rangle $ is an e-pair if:

• • $\mathcal {M}_1$ and $\mathcal {M}_2$ are adequate for e.

• • $\mathcal {M}_2$ is an $\in $ -extension of $\mathcal {M}_1$ .

• • $[a\in \mathcal {D}_2\mid \mathcal {M}_2\models N(a)]=[a\in \mathcal {D}_1\mid \mathcal {M}_1\models N(a)].$

Definition 4.21.

1. 1. A legal term t of $\mathbf {PW}$ is invariant if $\nu _{\mathcal {M}_1}[t]=\nu _{\mathcal {M}_2}[t]$ whenever $\langle \mathcal {M}_1,\mathcal {M}_2 \rangle $ is a t-pair, and $\nu $ is an assignment in $\mathcal {M}_1$ .

2. 2. A legal n-ary operation F of $\mathbf {PW}$ is invariant if

   $$ \begin{align*}F^{\mathcal{M}_1}[a_1 ,\ldots\!,\! a_n]=F^{\mathcal{M}_2}[a_1 ,\ldots\!,\! a_n]\end{align*} $$

   whenever $\langle \mathcal {M}_1,\mathcal {M}_2 \rangle $ is an F-pair, and $a_1 ,\ldots \!,\! a_n$ are elements of $\mathcal {M}_1$ .

3. 3. A legal formula $\varphi $ of $\mathbf {PW}$ such that $\{x_1 ,\ldots \!,\! x_n\}\subseteq Fv(\varphi )$ is invariant with respect to $\{x_1 ,\ldots \!,\! x_n\}$ if the following holds for every assignment $\nu $ in $\mathcal {M}_1$ :

   $$ \begin{align*}[\vec{a}: \mathcal{D}_2^n\mid \mathcal{M}_2,\nu\{\vec{x}:=\vec{a}\}\models\varphi]= [\vec{a}: \mathcal{D}_1^n\mid \mathcal{M}_1,\nu\{\vec{x}:=\vec{a}\}\models\varphi].\end{align*} $$

4. 4. A legal formula $\varphi $ of $\mathbf {PW}$ is persistent if $\mathcal {M}_1,\nu \models \varphi $ implies that ${\mathcal {M}_2,\nu \models \varphi }$ as well, whenever $\langle \mathcal {M}_1,\mathcal {M}_2 \rangle $ is a $\varphi $ -pair, and $\nu $ is an assignment in $\mathcal {M}_1$ .

Note 4.22. The definition of invariability of formulas can also be formulated as follows: a legal formula $\varphi $ of $\mathbf {PW}$ such that $Fv(\varphi )=\{x_1 ,\ldots \!,\! x_n,y_1 ,\ldots \!,\! y_k\}$ is invariant with respect to $\{x_1 ,\ldots \!,\! x_n\}$ if the following holds whenever $\langle \mathcal {M}_1,\mathcal {M}_2 \rangle $ is a $\varphi $ -pair, and $c_1, \ldots , c_k$ are elements of $\mathcal {M}_1$ :

$$ \begin{align*}[\vec{a}: \mathcal{D}_2^n\mid \mathcal{M}_2\models\varphi(\vec{a}^{\frown}\!\vec{c})]= [\vec{a}: \mathcal{D}_1^n\mid \mathcal{M}_1\models\varphi(\vec{a}^{\frown}\!\vec{c})]\end{align*} $$

(where $\langle a_1 ,\ldots \!,\! a_n \rangle ^{\frown }\!\langle c_1 ,\ldots \!,\! c_k \rangle = \langle a_1 ,\ldots \!,\! a_n,c_1 ,\ldots \!,\! c_k \rangle $ .)

Note 4.23. Identifying $\langle \rangle $ with $\emptyset $ , we get that in case $n=0$ , the collection $[\vec {a}: \mathcal {D}^n\mid \mathcal {M}\models \varphi (\vec {a}^{\frown }\!\vec {c})]$ is either 1 ( $=\{\emptyset \}$ ) or 0 ( $=\emptyset $ ), depending on whether $\mathcal {M}\models \varphi (\vec {c})$ or not. Hence invariance with respect to $\emptyset $ is simply absoluteness.

Convention. From now on, when we talk about terms, we shall mean legal terms. The same convention applies to formulas and operation symbols.

Theorem 4.24.

1. 1. Every term t of $\mathbf {PW}$ is invariant.

2. 2. Every operation F of $\mathbf {PW}$ is invariant.

3. 3. If $\varphi \succ \{x_1 ,\ldots \!,\! x_n\}$ in $\mathbf {PW}$ then $\varphi $ is invariant with respect to $\{x_1 ,\ldots \!,\! x_n\}$ .

4. 4. Every $\Sigma $ -formula of $\mathbf {PW}$ is persistent.

Proof We prove all parts simultaneously, using induction on the complexity of e, where e is a term or a formula or an operation symbol of $\mathbf {PW}$ . Nevertheless, to facilitate reading and understanding, we split the various induction steps into groups that correspond to the four parts of the theorem.

In what follows we assume that for every e we consider, $\langle \mathcal {M}_1,\mathcal {M}_2 \rangle $ is an e-pair ( $\mathcal {M}_1=\langle \mathcal {D}_1,I_1 \rangle $ , $\mathcal {M}_2=\langle \mathcal {D}_2,I_2 \rangle $ ), and $\nu $ is an assignment in $\mathcal {D}_1$ .

• Operations: Suppose that e is the n-ary operation symbol $F_{\varphi }$ . For convenience of presentation, assume that $n=1$ . Let $a:\mathcal {D}_1$ , and let $b=F_{\varphi }^{I_1}[a]$ . Since $\mathcal {M}_1$ is a model of $\mathbf {PW}_e$ , the legality of $F_{\varphi }$ and the rule [OpI] imply that $\mathcal {M}_1\models \varphi (\langle a,b \rangle )$ . Since $\varphi $ should be a $\Sigma $ -formula, the induction hypothesis for $\varphi $ implies that $\mathcal {M}_2\models \varphi (\langle a,b \rangle )$ too. Since $\mathcal {M}_2$ is a model of $\mathbf {PW}_e$ , this in turn implies (with the help of [OpI]) that $b=F_{\varphi }^{I_2}[a]$ as well.

• Terms:

  • • The case where e is a variable is trivial.

  • • The case $e=\omega $ is immediate from the third item of Definition 4.20.

  • • Suppose that e is the term $F_{\varphi }(s_1 ,\ldots \!,\! s_n)$ . Then

    $$ \begin{align*}\nu_{\mathcal{M}_1}[e]=F_{\varphi}^{I_1}[\nu_{\mathcal{M}_1}[s_1] ,\ldots\!,\! \nu_{\mathcal{M}_1}[s_n]]\ \ \ \nu_{\mathcal{M}_2}[e]=F_{\varphi}^{I_2}[\nu_{\mathcal{M}_2}[s_1] ,\ldots\!,\! \nu_{\mathcal{M}_2}[s_n]].\end{align*} $$

    Now from the induction hypothesis for $F_{\varphi }$ we get:

    $$\begin{align*}F_{\varphi}^{I_1}[\nu_{\mathcal{M}_1}[s_1] ,\ldots\!,\! \nu_{\mathcal{M}_1}[s_n]]= F_{\varphi}^{I_2}[\nu_{\mathcal{M}_1}[s_1] ,\ldots\!,\! \nu_{\mathcal{M}_1}[s_n]]\end{align*}$$

    while from the induction hypotheses for $s_1 ,\ldots \!,\! s_n$ we get that

    $$\begin{align*}F_{\varphi}^{I_2}[\nu_{\mathcal{M}_1}[s_1] ,\ldots\!,\! \nu_{\mathcal{M}_1}[s_n]]= F_{\varphi}^{I_2}[\nu_{\mathcal{M}_2}[s_1] ,\ldots\!,\! \nu_{\mathcal{M}_2}[s_n]].\end{align*}$$

    It follows that $\nu _{\mathcal {M}_1}[e]=\nu _{\mathcal {M}_2}[e]$ .

• The invariance relation: $\succ $

  • •

    • – To show the absoluteness (invariance with respect to $\emptyset $ ) of $s\in t$ , let $\nu $ be an assignment in $\mathcal {M}_1$ . Assume first that $\mathcal {M}_1,\nu \models s\in t$ . Then $\nu _{\mathcal {M}_1}[s]\in ^{I_1}\nu _{\mathcal {M}_1}[t]$ . It follows by the induction hypotheses for s and t and the fact that $\mathcal {M}_2$ is an $\in $ -extension of $\mathcal {M}_1$ that $\nu _{\mathcal {M}_2}[s]\in ^{I_2}\nu _{\mathcal {M}_2}[t]$ too. Hence $\mathcal {M}_2,\nu \models s\in t$ . The proof of the converse (i.e., that if $\mathcal {M}_2,\nu \models s\in t$ then $\mathcal {M}_1,\nu \models s\in t$ ) is similar.

    • – We leave the simpler proof that $s=t$ is absolute to the reader.

    • – The case $e=P_{\varphi }(t_1,\ldots \!,\! t_n)$ is immediate from the induction hypothesis for $\varphi $ , and the $\varphi $ -instance of [PrI] (which is in $\mathbf {PW}_e$ ).

  • •

    • – That $x\neq x$ is invariant with respect to x follows from the fact that for every $\mathcal {M}=\langle \mathcal {D},I \rangle $ and $\nu $ , $[a:\mathcal {D}\mid \mathcal {M},\nu \{x:=a\}\models x\neq x]=\emptyset $ .

    • – Let e be the formula $x= t$ , where $x\not \in Fv(t)$ . Then the collection $[a\in \mathcal {D}_1\mid \mathcal {M}_1,\nu \{x:=a\}\models x=t]$ is the singleton of $\nu _{\mathcal {M}_1}[t]$ , while $[a\in \mathcal {D}_2\mid \mathcal {M}_2,\nu \{x:=a\}\models x=t]$ is the singleton of $\nu _{\mathcal {M}_2}[t]$ . Hence the induction hypothesis for t implies that the two sets are equal.

    • – Let e be the formula $x\in t$ , where $x\not \in Fv(t)$ . Then

      $$ \begin{align*}[a\in\mathcal{D}_1\mid \mathcal{M}_1,\nu\{x:=a\}\models x\in t]=[a\in\mathcal{D}_1\mid a\in^{\mathcal{M}_1}\nu_{\mathcal{M}_1}[t]],\end{align*} $$
      $$ \begin{align*}[a\in\mathcal{D}_2\mid \mathcal{M}_2,\nu\{x:=a\}\models x\in t]=[a\in\mathcal{D}_2\mid a\in^{\mathcal{M}_2}\nu_{\mathcal{M}_2}[t]].\end{align*} $$
      Since $\nu _{\mathcal {M}_1}[t]=\nu _{\mathcal {M}_2}[t]$ by the induction hypothesis for t, these two equations and the fact that $\mathcal {M}_2$ is an $\in $ -extension of $\mathcal {M}_1$ imply:
      $$ \begin{align*}&[a\in\mathcal{D}_1\mid \mathcal{M}_1,\nu\{x:=a\}\models x\in t]\\&\quad=[a\in\mathcal{D}_2\mid \mathcal{M}_2,\nu\{x:=a\}\models x\in t].\end{align*} $$

  • • The proof that if $\varphi $ is absolute then so is $\neg \varphi $ is left to the reader.

  • • Let e be $\varphi \vee \psi $ , where $\varphi \succ \{x_1,\ldots \!,\! x_n\}$ and $\psi \succ \{x_1,\ldots \!,\! x_n\}$ . Then the collection $[\vec {a}\in \mathcal {D}_1^n\mid \mathcal {M}_1,\nu \{\vec {x}:=\vec {a}\}\models \varphi \vee \psi ]$ is the union of $[\vec {a}\in \mathcal {D}_1^n\mid \mathcal {M}_1,\nu \{\vec {x}:=\vec {a}\}\models \varphi ]$ and $[\vec {a}\in \mathcal {D}_1^n\mid \mathcal {M}_1,\nu \{\vec {x}:=\vec {a}\}\models \psi ]$ . A similar equation holds for $[\vec {a}\in \mathcal {D}_2^n\mid \mathcal {M}_2,\nu \{\vec {x}:=\vec {a}\}\models \varphi \vee \psi ]$ . The equality of these two collections follows therefore from the induction hypotheses for $\varphi $ and $\psi $ .

  • • Let e be $\theta =\varphi \wedge \psi $ , where $\varphi \succ X$ , $\psi \succ Y$ , and $Y\cap Fv(\varphi )=\emptyset $ . To simplify notation, assume that $Fv(\varphi )=\{x,z\}$ , $Fv(\psi )=\{x,y,z\}$ , $X=\{x\}$ , $Y=\{y\}$ . For $c\in \mathcal {D}_1$ , let $Z(c)=[a\in \mathcal {D}_2\mid \mathcal {M}_2\models \varphi (a,c)]$ . Since $\varphi \succ X$ , $Z(c)=[a\in \mathcal {D}_1\mid \mathcal {M}_1\models \varphi (a,c)]$ as well (by the induction hypothesis for $\varphi $ ). Hence $Z(c)\subseteq \mathcal {D}_1$ . By the induction hypothesis for $\psi $ , this and the fact that $\psi \succ Y$ imply that if $d\in Z(c)$ then

    $$ \begin{align*}[b\in \mathcal{D}_2\mid \mathcal{M}_2\models \psi(d,b,c)]= [b\in \mathcal{D}_1\mid \mathcal{M}_1\models \psi(d,b,c)].\end{align*} $$
    Denote this set by $W(c,d)$ . Now both $[\langle a,b \rangle \in \mathcal {D}_2^2\mid \mathcal {M}_2\models \theta (a,b,c)]$ and $[\langle a,b \rangle \in \mathcal {D}_1^2\mid \mathcal {M}_1\models \theta (a,b,c)]$ equal $\bigcup _{d\in Z(c)}\{d\}\times W(c,d)$ . Hence these two sets are the same (for every $c\in \mathcal {D}_1$ ).

  • • We leave to the reader the case $e=\exists y\varphi $ , where $\varphi \succ X$ and $y\in X$ .

• $\Sigma $ -formulas:

  • • Invariance implies persistence. Hence if $\varphi $ is absolute then it is persistent by what we have shown above.

  • • It is well known that persistence of formulas is closed under disjunction, conjunction, and existential quantification, so we leave to the reader the standard proofs of these cases.

  • • Suppose that $\theta $ is $\forall x(\varphi \to \psi )$ , where $\varphi \succ \{x\}$ and $\psi $ is a $\Sigma $ -formula. By the induction hypothesis for $\varphi $ and $\psi $ , $\varphi $ is invariant with respect to x, and $\psi $ is persistent. We show that $\theta $ is persistent too. So let $\nu $ be an assignment in $\mathcal {M}_1$ such that $\mathcal {M}_1,\nu \models \theta $ . We show that $\mathcal {M}_2,\nu \models \theta $ . So let $a\in \mathcal {D}_2$ . We should show that $\mathcal {M}_2,\nu \{x:=a\}\models \varphi \to \psi $ . This is certainly true in case $\mathcal {M}_2,\nu \{x:=a\}\not \models \varphi $ . So assume that $\mathcal {M}_2,\nu \{x:=a\}\models \varphi $ . Then the invariability of $\varphi $ with respect to x implies that $a\in \mathcal {D}_1$ , $\nu \{x:=a\}$ is an assignment in $\mathcal {D}_1$ , and $\mathcal {M}_1,\nu \{x:=a\}\models \varphi $ . Since $\mathcal {M}_1,\nu \models \theta $ , also $\mathcal {M}_1,\nu \{x:=a\}\models \varphi \to \psi $ . It follows that $\mathcal {M}_1,\nu \{x:=a\}\models \psi $ . This implies that $\mathcal {M}_2,\nu \{x:=a\}\models \psi $ , since $\psi $ is persistent. It follows that $\mathcal {M}_2,\nu \{x:=a\}\models \varphi \to \psi $ in this case too.

Clearly, Theorem 4.24 provides the predicative justification of [Comp] (according to the invariance criterion) that Theorem 4.12 does not. As for [OpI], its premise ensures that $F_{\varphi }$ can be introduced using the usual procedure of extension by definitions. That the resulting operation is invariant follows again from Theorem 4.24.

4.4 Comparison with Feferman’s systems

The design of $\mathbf {PW}$ has a lot in common with - and was greatly influenced by - the second-order system $P+\exists /P$ for predicative analysis that Feferman has developed in [Reference Feferman and Lorenz19]. In particular:

• • In practice, neither $\mathbf {PW}$ nor $P+\exists /P$ has a signature which is fixed in advance, and the method of repeatedly extending the language and adding new corresponding axioms is an essential component of the work in both.

• • The definitions of both systems involve a simultaneous recursive construction of their sets of symbols, terms, formulas, axioms, and rules.

• • The two special rules of $\mathbf {PW}$ , [OpI] and [Unif], which (as is shown below) give this system its strong power (far beyond that of the systems $\mathbf {RST}\omega $ and $\mathbf {PZF}$ investigated in [Reference Avron and Schindler5, Reference Avron and Cohen7, Reference Cohen and Avron10]), generalize (respectively) to set theory the following two rules from [Reference Feferman and Lorenz19]:

  • Functional defining rule: This rule allows to infer

    $$ \begin{align*}\vdash_P\forall v_1\forall v_0(\varphi(v_1,v_0)\leftrightarrow v_0=F_\varphi(v_1))\end{align*} $$
    from the premises $\vdash _P \forall v_1\forall v_2\forall v_3(\varphi (v_1,v_2)\wedge \varphi (v_1,v_3)\to v_2=v_3)$ and $\vdash _{\exists /P}\forall v_1\exists v_0\varphi (v_1,v_0)$ , provided that $\varphi $ is a formula in which quantification is made only over the lowest type N (implying that $\varphi $ is absolute).

  • Unification rule: This rule allows to infer

    $$ \begin{align*}\vdash_{\exists/P} \forall v_1\in\mathbf{N}\exists v_0\varphi(v_1,v_0)\to\exists f\forall v_1\in\mathbf{N}\varphi(v_1,f(v_1))\end{align*} $$
    from $\vdash _P \forall v_1\in \mathbf {N}\forall v_2\forall v_3(\varphi (v_1,v_2)\wedge \varphi (v_1,v_3)\to v_2=v_3)$ , provided that $\varphi $ is a formula in which quantification is made only over N.

  It should be noted that a preliminary version, called [F], of a combination of [OpI] and [Unif] appears already in the last section of [Reference Feferman15], where predicativity at higher types is discussed. [F] allows to infer from $\forall x\exists ! y \varphi (x,y)$ the formula $\exists f\forall x\forall y(\varphi (x,y)\leftrightarrow y=f(x))$ , provided that $\varphi $ is a $\Sigma $ -formula. (Note that unlike in the present context of type-free set theory, in [F] the variable f is of type higher than that of x.) The rule [F] was then implicitly split in [Reference Feferman and Lorenz19] into the two rules described above.

Nevertheless, there are also important differences between $\mathbf {PW}$ and ${P+\exists /P}$ :

1. 1. $\mathbf {PW}$ is a single system. In contrast, $P+\exists /P$ is a combination of two different ones: P and $\exists /P$ . P is taken to be the major system, while the stronger system $\exists /P$ is taken to be only an auxiliary one, which is needed for the precise definition of P. The connection between these two systems involves some choices for which no justification is given in [Reference Feferman and Lorenz19]. (See Weaver’s criticism, with which I fully agree, in Section 1.6 of [Reference Weaver50].) We believe, in fact (though we have not tried to show), that with different (but still predicatively justified) choices, $P+\exists /P$ can be strengthened to a single theory which is as strong as $\mathbf {PW}$ .

2. 2. P and $\exists /P$ are based on an extensive system of types, which has types of all finite levels (but actually uses only those of levels 0, 1, and 2). $\mathbf {PW}$ , in contrast, is a type-free, single-sorted set theory.

3. 3. Unlike $\mathbf {PW}$ , which is purely first-order, P and $\exists /P$ have second-order variables in addition to the first-order ones. In the case of $\exists /P$ it is even allowed (under certain conditions) to quantify on them. Moreover, both systems employ specific second-order rules, like substitution of terms, and even (again under certain conditions) of formulas, for second-order variables. (No rule of substitution is needed in the case of $\mathbf {PW}$ .)

4. 4. The set of natural numbers is taken as given in P. In $\mathbf {PW}$ it is defined.

Other systems of Feferman which are obviously related to $\mathbf {PW}$ are the pure set theories $\mathbf {PS_i}$ and $\mathbf {PS_iE}$ ( $i=0,1$ ) which were introduced in [Reference Feferman, Scott and Jech18] (see Note 1.4). Like $\mathbf {PW}$ , they too are intuitively motivated by the invariance criterion, and so $\Sigma $ -formulas and absolute formulas play an important role in their formulation. Accordingly, those systems are less restricted than the second-order systems of [Reference Feferman15]. Thus the union operation is allowed in them, although it is taken as impredicative in most of Feferman’s papers.Footnote ²¹ On the other hand, unlike $\mathbf {PW}$ , none of those systems respects our principle (I) from Section 1.4.2, since their axiomatizations include purely existential principles. An example is given by the $\Sigma $ -reflection rule, which allows to infer $\exists a\psi ^{(a)}$ from $\psi $ in case $\psi $ is a $\Sigma $ -formula (where $\psi ^{(a)}$ is obtained from $\psi $ by restricting each quantifier in it to a). $\mathbf {PS_iE}$ ( $i=0,1$ ) violates Principle (II) (from Section 1.4.2) as well, since it includes an axiom that says that every set is enumerable. Another very significant point of difference is due to the fact (shown in [Reference Feferman, Scott and Jech18]) that $\mathbf {PS_1}$ and $\mathbf {PS_1E}$ has a minimum model consisting of all sets constructible before $\Gamma _0$ . In contrast, the results of Section 7 imply than any transitive model of $\mathbf {PW}$ should contain $\Gamma _0$ . It follows that although the predicativity of the systems in [Reference Feferman, Scott and Jech18] is dubious, in some important sense they are all weaker than $\mathbf {PW}$ .Footnote ²²

5.1 Abstraction terms and RST

Let $Fv(\varphi )=\{x,v_1 ,\ldots \!,\! v_n\}$ ( $n\geq 0$ ), and suppose that $\varphi \succ \{x\}$ . Using [Comp], this entails

$$ \begin{align*}\vdash_{\mathbf{PW}}\forall v_1\dots\forall v_n\exists!v_0\forall x(x\in v_0\leftrightarrow \varphi).\end{align*} $$

Since the formula $\varphi ^\star :=\forall x(x\in v_0\leftrightarrow \varphi )$ is $\Sigma $ ,Footnote ²³ an application of [OpI] yields

$$ \begin{align*}\vdash_{\mathbf{PW}} F_{\varphi^\star}(v_1 ,\ldots\!,\! v_n)=v_0\leftrightarrow\forall x(x\in v_0\leftrightarrow \varphi(x,v_1 ,\ldots\!,\! v_n)).\end{align*} $$

Henceforth we shall write $\{x\mid \varphi (x,v_1 ,\ldots \!,\! v_n)\}$ instead of $F_{\varphi ^\star }(v_1 ,\ldots \!,\! v_n)$ (where $\varphi ^\star $ is defined from $\varphi $ as above), and call this an abstraction term. Obviously, $\vdash _{\mathbf {PW}} \forall x(x\in \{x\mid \varphi \}\leftrightarrow \varphi )$ . It follows that $\mathbf {PW}$ is an extension of the theory $\mathbf {RST}$ from [Reference Avron and Schindler5]. By the results of that paper, this implies that every rudimentary operation is definable in $\mathbf {PW}$ . Here are some examples of terms available in $\mathbf {RST}$ , and so in $\mathbf {PW}$ :

• • ${\emptyset }=_{Df}\{x\mid x\neq x\}.$

• • ${\{t_1 ,\ldots \!,\! t_n\}}=_{Df}\{x\mid x=t_1\vee \dots \vee x=t_n\}.$

• • ${\langle t,s \rangle }=_{Df}\{\{t\},\{t,s\}\}.$

• • ${\{x\in t\mid \varphi \}}=_{Df}\{x\mid x\in t\wedge \varphi \}$ (provided $\varphi \succ \emptyset $ and $x\not \in Fv(t)$ ).

• • ${\{t(x)\mid x\in s\}}=_{Df}\{y\mid \exists x.x\in s\wedge y=t\}.$

• • ${\bigcup t}=_{Df}\{x\mid \exists y.y\in t\wedge x\in y\}.$

• • ${s\times t}=_{Df}\{x\mid \exists a \exists b. a\in s \wedge b\in t \wedge x=\langle a,b \rangle \}.$

• • ${\iota x\varphi }=_{Df}\bigcup \{x\mid \varphi \}$ (provided $\varphi \succ \{x\}$ ).

• • ${\lambda x\in s.t}=_{Df} \{\langle x,t \rangle \mid x\in s\}.$

• • ${f(x)}=_{Df} \iota y.\exists z\exists v(z\in f\wedge v\in z\wedge y\in v\wedge z=\langle x,y \rangle ).$

• • $Dom(f)=\{x\in \bigcup f\mid \exists y\in \bigcup \bigcup f.\langle x,y \rangle \in f\}.$

• • $Im(f)=\{x\in \bigcup f\mid \exists y\in \bigcup \bigcup f.\langle y,x \rangle \in f\}.$

• • $f\upharpoonright s=_{Df}\{\langle x,f(x) \rangle \mid x\in s\}$ (where x is new).

The following are examples of easy related theorems of RST:

• • $\exists ! x\varphi (x)\to \forall x(\varphi (x)\leftrightarrow x=\iota x\varphi (x))$ (if $\varphi \succ \{x\}$ ).

• • $u\in s\to (\lambda x\in s.t)u=t\{u/x\}$ (if u is free for x in t).

• • $Fun(f)\to (\langle x,y \rangle \in f\leftrightarrow y=f(x))$ , where $Fun(f)$ is the following absolute formula (which says that f is a function):

  $$ \begin{align*}\forall z\in f\exists x\exists y(z=\langle x,y \rangle)\wedge\forall x\forall y_1\forall y_2( \langle x,y_1 \rangle\in f\wedge \langle x,y_2 \rangle\in f\to y_1=y_2).\end{align*} $$

5.2 Explicit definitions and the extended [OpI]

5.2.2 The extended [OpI]

The use of [OpI] can be made more effective with the help of the following proposition:

Proposition 5.3. For $1\leq i\leq k$ , let $\varphi _i$ and $\psi _i$ be formulas of $\mathbf {PW}$ such that $\varphi _i\in \Sigma $ , $Fv(\varphi _i)=\{v_0,v_1,\ldots \!,v_n\}$ , $\psi _i\succ \emptyset $ , and $Fv(\psi _i)\subseteq \{v_1 ,\ldots \!,\! v_n\}$ . Suppose that for every $1\leq i,j\leq k$ such that $i\neq j$ we have that $\vdash _{\mathbf {PW}}\neg (\psi _i\wedge \psi _j)$ and that $\vdash _{\mathbf {PW}}\forall v_1 ,\ldots \!,\! \forall v_n(\psi _i\to \exists !v_0\varphi _i)$ . Then $\mathbf {PW}$ has an operation F such that for every $1\leq i\leq k$ : $\vdash _{\mathbf {PW}}\forall v_1\dots \forall v_n(\psi _i\to \varphi _i\{F(v_1 ,\ldots \!,\! v_n)/v_0\})$ .

Proof $F\!=\!F_\theta $ , where $\theta \!=\! (\psi _1\wedge \varphi _1)\!\vee \cdots \vee \!(\psi _k\wedge \varphi _k)\!\vee \! (\neg (\psi _1\wedge \dots \psi _k)\wedge v_0=0)$ .

We shall call the principle stated in Proposition 5.3 the extended [OpI].

Example 5.4. Let $\varphi $ be the following $\Sigma $ -formula (where we write $y,x,n$ instead of $v_0,v_1,v_2$ , respectively).

$$ \begin{align*}\exists f(Fun(f)\wedge Dom(f)=S(n) \wedge f(0)=x\wedge f(n)=y\wedge \forall i\in n.f(i+1)=\bigcup f(i).\end{align*} $$

An induction on n shows that $\vdash _{\mathbf {PW}}\forall x\forall n(n\in \omega \to \exists !y\varphi )$ . Hence the extended [OpI] implies that $\vdash _{\mathbf {PW}}\forall x\forall n(n\in \omega \to \varphi \{H(x,n)/y\})$ for some operation H of $\mathbf {PW}$ . Let $RTC(x)=\cup \{H(n,x)\mid n\in \omega \}$ . Then $RTC(x)$ represents in $\mathbf {PW}$ the reflexive–transitive closure of x, and so $TC(x)=RTC(x)-\{x\}$ represents the transitive closure of x. Hence the relation $\in ^\star $ (the transitive closure of $\in $ ) is definable in $\mathbf {PW}$ by $y\in ^\star x=_{Df}y\in TC(x)$ .

5.3 Transitive closures and PZF

Let $\psi $ be a formula of $\mathbf {PW}$ such that $\{x,y\}\subseteq Fv(\psi )$ , and $\psi \succ \{y\}$ . Define $\varphi $ like in Example 5.4, replacing the conjunct $\forall i\in n.f(i+1)=\bigcup f(i)$ by $\forall i\in n.f(i+1)=\{y\mid \exists z\in f(i).\psi \{z/x\}\}$ . Following exactly the same procedure as in Example 5.4, we construct in $\mathbf {PW}$ an operation $TC_\psi $ such that $TC_\psi (x,y):=y\in TC_\psi (x)$ is true iff there is a finite $\psi $ -chain that connects y to x. Hence this formula is semantically equivalent to the formula $(TC_{x,y}\psi )(x,y)$ of PZF. Moreover, like the latter, $TC_\psi (x,y)\succ \{y\}$ . It is also easy to prove in $\mathbf {PW}$ that $TC_\psi (x,y)$ has all the properties that $(TC_{x,y}\psi )(x,y)$ has in PZF according to Section 3.3 of [Reference Avron and Schindler5]. It follows that $\mathbf {PW}$ contains PZF.

5.4 The use of $\Delta $ -formulas

Proof Let $\psi :=(\varphi \wedge v_0=1)\vee (\neg \varphi \wedge v_0=0)$ . Obviously, $\vdash _{\mathbf {PW}}\forall v_1\dots \forall v_n\exists !v_0\psi $ , and $\psi $ is equivalent in $\mathbf {PW}$ to a $\Delta $ -formula $\psi ^\star $ . Therefore by [OpI] we get

$$ \begin{align*}\vdash_{\mathbf{PW}}\forall v_1\dots\forall v_n(\varphi\wedge F_{\psi^\star}(v_1 ,\ldots\!,\! v_n)=1) \vee(\neg\varphi\wedge F_{\psi^\star}(v_1 ,\ldots\!,\! v_n)=0).\end{align*} $$

It follows that $\vdash _{\mathbf {PW}}\varphi \leftrightarrow F_{\psi ^\star }(v_1 ,\ldots \!,\! v_n)=1$ . Let $\varphi ^\star :=F_{\psi ^\star }(v_1 ,\ldots \!,\! v_n)=1$ . Then $\varphi ^\star $ is absolute. Hence we may apply [PrI] to it. Take $R_{\varphi }$ to be $P_{\varphi ^\star }$ .

Corollary 5.8. Let $\varphi $ be a $\Delta $ -formula, $Fv(\varphi )=\{v_1 ,\ldots \!,\! v_n\}$ . Then:

$$ \begin{align*}\vdash_{\mathbf{PW}}\forall a \exists Z\forall v_1\dots\forall v_n(\langle v_1 ,\ldots\!,\! v_n \rangle\in Z\leftrightarrow\langle v_1 ,\ldots\!,\! v_n \rangle\in a\wedge\varphi).\end{align*} $$

5.5 Predicative set-theoretic recursion

Theorem 5.10. Let F be an ( $n+2$ )-ary operation of $\mathbf {PW}$ . Then $\mathbf {PW}$ has an ( $n+1$ )-ary operation G such that

$$ \begin{align*}\vdash_{\mathbf{PW}} \forall z_1\dots\forall z_n\forall x.G(z_1 ,\ldots\!,\! z_n,x)=F(z_1 ,\ldots\!,\! z_n,x,G\!\upharpoonright\!TC(x))\end{align*} $$

(where by $G\!\upharpoonright \!y$ we mean the function $\lambda u\in y.G(z_1 ,\ldots \!,\! z_n,u)$ ).

Proof For simplicity of the presentation, we prove the case $n=0$ . We freely use (here and later) facts about provability in $\mathbf {PW}$ that can easily be seen.

Define a formula $\varphi $ such that $Fv(\varphi )=\{x,f\}$ by

$$ \begin{align*}\varphi\kern1.2pt{:=}\kern1.2ptFun(f)\kern1.2pt{\wedge}\kern1.2pt Dom(f)\kern1.2pt{=}\kern1.2pt RTC(x)\wedge\forall z\kern1.2pt{\in}\kern1.2pt RTC(x). f(z)\kern1.2pt{=}\kern1.2pt F(z,f\!\upharpoonright\!TC(z)).\end{align*} $$

Next we show that $\varphi $ has the following properties.

1. 1. $\varphi $ is absolute.This easily follows from Example 5.4.

2. 2. $\vdash _{\mathbf {PW}}\varphi (x,f)\wedge z\in RTC(x)\to \varphi (z,f\!\upharpoonright \!RTC(z)).$

   This follows from $\vdash _{\mathbf {PW}} z\in x\to RTC(z)\subseteq RTC(x)$ and $\varphi $ ’s definition.

3. 3. $\vdash _{\mathbf {PW}}\varphi (x,f_1)\wedge \varphi (x,f_2)\to f_1=f_2.$

   The proof is by $\in $ -induction on x, using the previous item and the fact that $\vdash _{\mathbf {PW}} RTC(x)=\{x\}\cup \bigcup \{RTC(z)\mid z\in x\}$ .

4. 4. $\vdash _{\mathbf {PW}}\varphi (x_1,f_1)\wedge \varphi (x_2,f_2)\wedge z\in RTC(x_1)\cap RTC(x_2) \to f_1(z)=f_2(z).$

   This follows from the previous two items and the definition of $\varphi $ .

5. 5. $\vdash _{\mathbf {PW}}\forall x\exists f\varphi (x,f).$

   The proof is by an $\in $ -induction (in $\mathbf {PW}$ ) on x. So assume (in $\mathbf {PW}$ ) that $\forall z\in TC(x) \exists g\varphi (z,g)$ . Using an application of [Unif] (which is justified by items 1 and 3), it follows from this assumption that there is a function g such that $Dom(g)=TC(x)$ and $\varphi (z,g(z))$ holds for every $z\in TC(x)$ . Let $f^\star =\bigcup \{g(z)\mid z\in TC(x)\}$ . From item 4 it follows that $f^\star $ is a function whose domain is $TC(x)$ . Let $f=f^\star \cup \{\langle x,F(x,f^\star ) \rangle \}$ . It is straightforward to verify that $\varphi (x,f)$ .

Define $\psi (x,y)=\exists f.\varphi (x,f)\wedge f(x)=y$ . From items 3 and 5 it follows that $\vdash _{\mathbf {PW}}\forall x\exists !y\psi (x,y)$ , while item 1 implies that $\psi $ is a $\Sigma $ -formula. Therefore by applying [OpI] we get an operation G such that $\vdash _{\mathbf {PW}}\psi \{G(x)/y\}$ . Obviously, G has the required property.

6.1 Basic theory of ordinals

As usual, we use small Greek letters to vary over ordinals (writing, e.g., $\exists \alpha \varphi $ instead of $\exists x(On (x) \wedge \varphi )$ and $\forall \alpha \varphi $ instead of $\forall x(On(x)\to \varphi )$ ). We shall also frequently write $\alpha <\beta $ instead of $\alpha \in \beta $ and $\alpha \leq \beta $ instead of $\alpha \in \beta \vee \alpha =\beta $ .

From now on, we leave to the readers most of the proofs of claims about provability in $\mathbf {PW}$ in case the proofs in $\mathbf {PW}$ are practically just the standard ones, and their availability in $\mathbf {PW}$ can easily be checked. (This does not include, of course, any proof which makes use of [PrI], [OpI], or [Unif].)

6.2 Operations on ordinals