1.1. Simple examples

The Wand/Set Template is very abstract. So it will help to start with some concrete instances of the Template. This will also illustrate the scope of my Main Theorem.

Example 1.1 (Conway-games)

Conway [Reference Conway7] famously provided a general theory of (hereditary) two-player games. A Conway-game has left-options and right-options, and Conway-games are identical iff they have the same left- and right-options.

Here is a sketch of how to provide a wand/set theory of Conway-games. We have a single wand, game. Tapping an ordered-pair, ${\langle a,b\rangle }$ , of bland sets with game (typically) yields a new object; we deem this to be a Conway-game whose left-options are exactly a’s members and whose right-options are exactly b’s members. By courtesy, a bland set, a, is the Conway-game with no right-options and whose left-options are exactly a’s members.

Here is the idea in a bit more detail. Reserve curly-brackets for bland sets, and $\in $ for membership of the same. With , stipulate that game only acts on objects of the form ${\langle a,b\rangle }$ , where both a and b are bland sets. Then , and if $b \neq \emptyset $ then $\mathsf{game}({\langle a,b\rangle })$ is an object which is not a bland set; and games are governed by this rule:

$$ \begin{align*}\mathsf{game}({\langle a,b\rangle}) = \mathsf{game}({\langle a',b'\rangle}) \rightarrow (a = a' \land b = b').\end{align*} $$

Reading “” as “x is a left-option of y”, and similarly for “”, explicitly define:

Properly implemented, we have a theory whose objects are exactly the Conway-games. By my Main Theorem, this is synonymous with a ZF-like theory.Footnote ¹

1.2. Conway’s liberationism

Further instances of the Wand/Set Template could be offered, but I hope that Examples 1.1–1.4 already illustrate the Template’s power. I now want to move beyond piecemeal examples and explore a more principled reason for investigating the Template.

Immediately after outlining his theory of games (see Example 1.1), Conway advocated for a Mathematicians’ Liberation Movement. His liberationism held that:

1. (i) “Objects may be created from earlier objects in any reasonably constructive fashion.

2. (ii) Equality among the created objects can be any desired equivalence relation.”Footnote ⁹

He also expected that there would be a meta-theorem, which would show that any theory whose objects are created in such a “reasonably constructive” fashion can be embedded within (some extension of) ZF.

As the examples of Section 1.1 suggest, my Wand/Set Template realizes Conway’s proposed liberationism. Regarding (i): to implement the idea of creating some object “from earlier objects”, all you need to do is form their bland set, and then tap that set with an appropriate wand (or perhaps with the right wands in the right order). Regarding (ii): when we specify the behaviour of our wands, we can allow “any desired equivalence relation” to dictate when the result of tapping a with wand $w$ is equal to the result of tapping b with wand u. And my Main Theorem exceeds Conway’s expectations: all loosely constructive implementations of the Wand/Set Template are not merely embeddable in (some extension of) ZF, but synonymous with a ZF-like theory.

Admittedly, my Template may not fully exhaust the scope of Conway’s liberationism. This is because Conway’s (i) and (ii) give us permissions, not obligations; they say what we may create, without dictating what we must create. My Template, by contrast, insists on creatingFootnote ¹⁰ bland sets at each stage. However, my insistence is unlikely to impinge seriously on Conway’s proposed liberationism. For one thing: whenever Conway creates entities of some kind, K, he is confident that they will be embeddable into a ZF-style hierarchy; so it is no great burden to request that, when he creates the Ks, he also create the ZF-like bland-sets into which his Ks will be embedded. For another: if we really want our theory to be a theory of Ks only, we can often do this by creatively (and courteously) regarding bland sets as Ks of a particular sort: we saw this in Examples 1.1–1.3, though not in Example 1.4.

1.3. Church’s set theory with a universal set

I have presented some examples of the Wand/Set Template, and explained how the Template relates to Conway’s Liberationism. I now want to consider a more surprising connection: we can regard Church’s [Reference Church and Henkin6] set theory with a universal set as both an instance of the Wand/Set Template and as an exercise of Conway’s liberationism.Footnote ¹¹

Church’s [Reference Church and Henkin6] idea is roughly this. We have a ZF-like hierarchy of bland sets. We also have some wands: a complement wand and, for each positive natural n, a cardinal $_n$ wand. These wands should behave as follows:

1. (0) Tapping any object, a, with complement yields a’s absolute complement; that is, the set whose members are exactly the things not in a.

2. (n) Tapping some bland set, a, with cardinal $_n$ , yields the set of all bland sets n-equivalent with a (if there are any; if there are none, this yields nothing; and tapping anything other than a bland set with $\mathsf{cardinal}_n$ yields nothing).

This explanation mentions n-equivalence. This is an explicitly defined equivalence relation (defined recursively on n; see Definition 11.1). Its details are largely irrelevant for now; what matters is just that Church’s theory can be thought of as invoking a countable infinity of wands with explicitly defined actions.

What is particularly interesting about Church’s theory, however, is that—unlike Examples 1.1–1.4—it does not obviously seem like an instance of Conway’s liberationism. Conway’s (ii) says that “Equality among the created objects can be any desired equivalence relation”; this is fine, since Church’s wands are based around explicitly defined equivalence relations. But Conway’s (i) says that “Objects may be created from earlier objects in any reasonably constructive fashion”; Church’s wands seem to break this condition. After all: following Church, tapping $\emptyset $ with complement must reveal the universal set, V; but things we have not yet discovered must be members of V; so one might well doubt whether this is “reasonably constructive”!

Crucially, this doubt can be addressed. The complement of a bland set is never itself bland. So, where $\in $ is the primitive membership relation we use for bland sets, we can explicitly define an extended sense of “membership” which is suitable for both bland sets and their complements, along these lines:

$$ \begin{align*} x \mathrel{\varepsilon} y &\mathrel{\mathord{:}\mathord{\equiv}} ({Bland}(y)\land x \in y) \lor (\exists c : {Bland})(y =\mathsf{complement}(c) \land x \notin c). \end{align*} $$

Now, $\emptyset $ is bland, and nothing is in $_{\mathrel {\varepsilon }}$ it; and $\emptyset $ ’s complement, V, is not bland, and everything is in $_{\mathrel {\varepsilon }}$ it. By definition, though, the latter claim is exactly as constructive as the former. Doubt resolved!

Of course, this is only a quick sketch of how to make Church’s [Reference Church and Henkin6] theory look “reasonably constructive”: it says nothing about the cardinal $_n$ wands, and omits a lot of other details. Still, it gives us all the right ideas, and the rest really is just details (for which, see Section 11). The surprising upshot is that Church’s theory can be regarded as a loosely constructive implementation of the Wand/Set Template. By my Main Theorem, it is therefore synonymous with a ZF-like theory.Footnote ¹²

Having tackled Church’s theory, it is natural to ask whether we can grapple with other theories which are—at least at first blush—deeply non-constructive. Specifically, we might observe that Quine’s NF can be given a finite axiomatization which is based on the idea that there are certain “starter” sets, and that the other sets are formed from these by simple operations, such as taking complements.Footnote ¹³ Regarding the operations as wand-actions, we might then hope to be able to regard NF itself as an implementation of the Wand/Set Template. This might even lead to a consistency proof for NF, or—as Church suggested—a “synthesis or partial synthesis of” ZF with NF.Footnote ¹⁴

Clearly, the Wand/Set Template opens up some giddying possibilities. Inspired by them, my aim is to set the Template on a formal footing, and then establish my Main Theorem: any loosely constructive way of fleshing out the Wand/Set Template is synonymous with a ZF-like theory. (For better or worse, this Theorem ultimately restores some sobriety to our giddy state, since it seems to undermine the possibility of saying much about NF using the Wand/Set Template.Footnote ¹⁵ )

I will explain this synonymy result more precisely as I go. After some simple preliminaries (Sections 2 and 3), I will explicate what it takes to flesh out the Wand/Set Template in a loosely constructive way (Sections 4 and 5). This will allow me to present a general definition of an arbitrary wand/set theory (Definition 5.1). I will then explain how any wand/set theory mechanically generates a theory which is “ZF-like” in a precise sense (Section 6). Having done this, I will prove the Main Theorem (Sections 7–10). I conclude by applying my wand/set apparatus to Church’s [Reference Church and Henkin6] theory (Section 11).

4.1. The Picture

I will start with some very soft imagery. Imagine we are about to tap some object, a, with a wand, $w$ , for the first time. We are trying to calculate what, if anything, we will find as a result of this wand-tap, i.e., whether will exist and what it will be like. Now: what resources should we be allowed to invoke in our calculations?

Evidently, we should be allowed to refer to any objects which we have already found. They are, after all, lying around for us to inspect. But if we attempt to consider objects which have not yet been found, we start to run the risk of vicious circularity. After all, suppose we think that whether will exist depends upon which objects will exist later. Well, what exists later depends upon whether will exist after the wand-tap we are about to perform, and we have run into circularity.

Such concerns are completely familiar from discussions of (im)predicativity in set theory. But they need not force us to adopt a fully predicative standpoint. After all, most of us are perfectly relaxed about the amount of impredicativity licensed by ZF. So I propose that we be allowed exactly that amount of impredicativity.

Let me spell out what this proposal amounts to. Our Template tells us that we find new bland sets at every stage. Moreover, these bland sets behave just like ZF-style sets. So, whatever objects we have already found (via whatever esoteric use of wands), we know that there will be a “bland-hierarchy” which is built from them, intuitively treating the already-found objects as urelements and collecting them together into (bland) sets without any further deployment of our wands. This “bland-hierarchy” is exactly as impredicative as ordinary ZF; so we should be allowed to make reference to it, but nothing outside it.

Since things are getting complicated, a picture may help. (Compare this with the big “V” we draw to describe the “ordinary” set-theoretic hierarchy.)

The unshaded area represents what we have already found; we are allowed to refer to this part of the hierarchy in our calculations, since it is perfectly transparent to us. The lightly shaded area represents the bland-hierarchy constructed from the earlier-found objects. Although this bland-hierarchy is undiscovered as yet, its impredicativity is exactly of a piece with that of ordinary ZF; we are allowed to refer to this part of the hierarchy in our calculations, since its behaviour has already been tamed. But the darkly shaded area represents objects we will find (only) by future uses of wand-taps; we are barred from referring to this part of the hierarchy in our calculations since, as legend warns, here be viciously circular dragons.

In what follows, I refer to this image as the Picture of Loose Constructivism. This Picture guides me through the next couple of sections. But my immediate task is to make this (informal) Picture more precise, so that we can embed it within our formal theorizing.

4.2. Loosely constructive formulas

I start by explicitly defining the “bland-hierarchy built from $\text {u}$ ”, where $\text {u}$ is any set of objects which we will treat as urelements. Intuitively, we treat $\text {u}$ as given; we then resist the temptation to deploy any wands, but keep repeatedly forming all possible bland sets.

Evidently, this “bland-hierarchy” should be arranged into well-ordered layers, which we might call levels $_{\text {u}}$ . Indeed, writing ${V}^{\alpha }_{\text {u}}$ for the $\alpha \text {th}$ level $_{\text {u}}$ , a little reflection on the “intuitive” idea just sketched indicates that the levels $_{\text {u}}$ should behave thus:Footnote ²³

$$ \begin{align*} {V}^{\alpha}_{\text{u}} &= \{x \mathrel{:} {Bland}(x) \land (\exists \beta < \alpha)x \subseteq {V}^{\beta}_{\text{u}}\} \cup \text{u.} \end{align*} $$

This identity would work perfectly well as a recursive definition of the levels $_{\text {u}}$ .Footnote ²⁴ However, it will be slightly easier for us if we instead, officially, define the levels $_{\text {u}}$ non-recursively (noting that this is equivalent to the recursive approach, by Lemma 7.12):Footnote ²⁵

Definition 4.1. For any $\text {u}$ , say that:Footnote ²⁶

$$ \begin{align*} x \lhd_{\text{u}} a&\mathrel{\mathord{:}\mathord{\equiv}} \big({Bland}(x) \land \exists c(x \subseteq c \in a)\big) \lor x \in \text{u.} \end{align*} $$

Let . Say that ${\textit{Hist}}_{\text {u}}(h)$ iff . Say that t is a level $_{\text {u}}$ , written ${\textit{Lev}}_{\text {u}}(t)$ , iff . Say that $\textbf {V}_{\text {u}}(a)$ iff $(\exists t : {\textit{Lev}}_{\text {u}})a \in t$ .

Now the formal claim that x satisfies $\textbf {V}_{\text {u}}$ precisely explicates the intuitive idea that x falls somewhere in the “bland-hierarchy built from $\text {u}$ ”.

I can now explain how to embed the Picture of Loose Constructivism within our formal theorizing. Suppose we are at stage s, and some formula, , is supposed to tell us something about what will happen later. So must deploy (only) resources which count as loosely constructive at s. In this case, I will say that is loosely bound by s; and I will now define this notion precisely.

Given the Picture, will be loosely bound by s iff considers (only) the bland-hierarchy built from the objects found at s, i.e., the bland-hierarchy built from the objects in $s^+$ , where $s^+$ is the level immediately after s.Footnote ²⁷ So should range (only) over $\textbf {V}_{s^+}$ . Moreover, if mentions any wand-taps, i.e., if contains expressions of the form “”, then, since is never bland (by InNB), must contain a guarantee that is found no later than s; this is equivalent to insisting that x itself is found strictly before s,Footnote ²⁸ i.e., to insisting that $x \in s$ .

These ideas are rolled together in the following definition.

To round off this discussion, it is worth giving particular consideration to the bland-hierarchy which is built by treating nothing as an urelement, i.e., the denizens of $\textbf {V}_{\emptyset }$ . These are bland sets, whose members are bland, whose members of members are bland … and so on, all the way down. In a word, they are hereditarily bland. These are important because, given the Picture, hereditarily bland sets are precisely the objects which are available to consider “at the outset”. It turns out (Lemma 7.15) that we can equivalently define hereditarily bland sets without mentioning $\textbf {V}_{\emptyset }$ , and this will be my official definition:

5.1. The wands are hereditarily bland

I have said that any wand/set theory will need to have the three quasi-notational axioms (InNB, TapNB, and TapFun from Section 2) and the three axioms which arrange everything into well-ordered wevels (Ext, Sep, and Strat from Section 3). But I have said nothing, yet, about the wands. This obviously needs to addressed.

Different instances of the Template will have different wands. Still, we can still make a general comment. The Template tells us to tap everything we can, at every stage, with every wand. This means that every wand must be “available” to us at the outset. But only hereditarily bland sets are “available” to us at the outset (see the end of Section 4.2). So every wand must be hereditarily bland. We therefore adopt the axiom:

• HebWands $(\forall w : Wand){Heb}(w).$

5.2. Weak claims about the hierarchy’s height

Next, I lay down an axiom which makes a weak claim about the hierarchy’s height:

• WeakHeight $\{w \mathrel {:} Wand(w)\}$ exists; and for every ordinal $\alpha $ both $\alpha \cup \{\alpha \}$ and $\operatorname{R}(\{w \mathrel {:} Wand(w)\}) + \alpha $ exist.Footnote ²⁹

Given the ambient axioms, to say that $\{w \mathrel {:} Wand(w)\}$ exists is to say that the hierarchy is sufficiently tall that all the wands are (eventually) found together at some wevel. The statement about an ordinal-sum existing just makes a further claim about the hierarchy’s height. After all, each ordinal $\beta $ is first found at the $\beta \text {th}$ wevel (see Section 3). So we can control the hierarchy’s height just by insisting that certain ordinals exist; and that is what WeakHeight does.

Having understood roughly what WeakHeight says, we need to ask why we should adopt it. Candidly, WeakHeight is proof-generated: it is the weakest height-principle which licenses my proof of synonymy!Footnote ³⁰ I concede that it goes beyond anything mentioned in the Wand/Set Template—the Template itself allows that we might have precisely forty-two stages, for example—but I note that it really is extraordinarily weak for most mathematical purposes.Footnote ³¹ I adopt it going forward.

5.3. Variable axioms for hereditarily bland sets

So far, I have specified some generic axioms which will hold for any wand/set theory. I now need to start considering axioms which vary between theories.

To illustrate: Example 1.1 only required a single wand (see Section 1.1), but Church’s [Reference Church and Henkin6] theory treated $\omega $ as the set of wands (see Section 1.3). Saying that $\omega $ exists makes a specific demand on the height of the hierarchy, which is independent of the axioms we have laid down so far. But it is a demand which, with Church, we may well want to make.

We can make demands on the height of the hierarchy in other ways too. For example: suppose we add ZF’s Replacement scheme, restricting all the quantifiers in each scheme-instance to hereditarily bland sets; via perfectly familiar reasoning, this will ensure that our hierarchy is strongly inaccessible.

We might want to say this; we might not; it all depends on exactly how we want to implement the Wand/Set Template. In order to be maximally accommodating, I allow the following. A wand/set theory can take any further sentences as axioms, provided that they are the result of taking a sentence, , in the signature $\{Wand, \in \}$ , and then restricting all of ’s quantifiers to hereditarily bland sets. (In what follows, I use the notation for this.)

Such axioms allow us to characterize the height of the hierarchy, precisely by characterizing the height of its hereditarily bland inner model. And any axiom which has been laid down in this way automatically conforms with loose constructivism. After all, the hereditarily bland sets are all available at the outset (see the end of Section 4.2).

Such axioms will also allow us to say that the set of wands is $\omega $ , or the least strongly inaccessible ordinal, or what-have-you. Of course, such axioms might lead us to inconsistency; there is no foolproof method for guarding against that. But nor should we expect such a method, any more than we expect one when we consider extensions of ZF.

5.4. Axioms for wand-taps

Finally, we need some axioms which tell us how wand-taps behave. Different wand/set theories will, of course, say different things at this point; but we can still offer some useful general principles.

Nothing I have said yet tells us whether (and when) we should find some object by tapping a with $w$ , i.e., whether exists. We will address this by asking whether a is in w’s “domain of action”, written ${Dom}(w/a)$ . Of course, different theories will specify different “domains” for different wands. So, to specify a wand/set theory, we must explicitly define some formula, ${Dom}(w/a)$ , with all free variables displayed, and we then stipulate that ${Dom}$ is a necessary and sufficient condition for finding something by wand-tapping:

• Making ${Dom}(w/a) \leftrightarrow \exists c\ \textit{Tap}(w/a,c).$

We also need to know whether (and when) entities found by wand-tapping are identical, i.e., whether . We will address this by asking whether $w$ and a (in that order) are “equivalent” to u and b (in that order); we write this ${Eq}(w/a,u/b)$ . Of course, different theories will use different notions of equivalence. So, to specify a wand/set theory, we must explicitly define some formula, ${Eq}(w/a,u/b)$ , with all free variables displayed, and we then lay down a kind of quotienting principle:

• Equating $\textit{Tap}(w/a,c) \rightarrow \big (\textit{Tap}(u/b,c) \leftrightarrow {Eq}(w/a,u/b)\big ).$

Otherwise put: if exists, then iff $w,a$ is equivalent to $u,b$ .

As remarked: the axioms Making and Equating invoke make use of formulas, ${Dom}$ and ${Eq}$ , which a wand/set theory must explicitly define. So the last question is: What sorts of definitions are allowed? Unsurprisingly, I want to maximize flexibility. So I want to say that any definitions are allowed, provided they meet these six constraints:

• (dom1) ${Dom}(w/a)$ entails $Wand(w)$ ;

• (dom2) ${Dom}(w/a)$ is loosely bound by ${\mathbb {L}{a}}$ ;

• (dom3) ${Dom}$ is preserved under equivalence, in that we always have:

  $$ \begin{align*} ({Eq}(w/a,u/b) \land {Dom}(w/a)) &\rightarrow {Dom}(u/b); \end{align*} $$

• (eq1) ${Eq}(w/a,u/b)$ entails $Wand(w)$ and $Wand(u)$ ;

• (eq2) ${Eq}(w/a,u/b)$ is loosely bound by ${\mathbb {L}{a}} \cup {\mathbb {L}{b}}$ ;

• (eq3) ${Eq}$ describes an equivalence relation, in that we always have:

  $$ \begin{align*} Wand(w) &\rightarrow {Eq}(w/a,w/a),\\ ({Eq}(w/a,u/b) \land {Eq}(w/a,v/c)) &\rightarrow {Eq}(u/b,v/c). \end{align*} $$

These six constraints really just make explicit some of our implicit commitments. We insist on (dom1) and (eq1) because we are considering tapping things with wands, and only with wands. The Picture of Loose Constructivism forces (dom2) and (eq2) upon us: these constraints ensure that, when we (first) ask whether we should find anything by tapping a with $w$ , or whether $w,a$ is equivalent to $u,b$ , we offer answers using only resources which are “available” to us there and then. The need for (eq3) is wholly obvious: we want ${Eq}$ to define a kind of equivalence, which governs identity between wand-taps via Equating. The most interesting constraint is (dom3). To see why we need it: suppose that ${Dom}(w/a)$ , so that tapping should indeed exist, by Making. Suppose also that $w,a$ is equivalent to $u,b$ ; then by Equating. But then, by Making, we must also have that ${Dom}(u/b)$ .

This explains why I have imposed conditions (dom1)–(dom3) and (eq1)–(eq3); it remains to explain how I will impose them.

Intuitively, the idea is as follows. We can lay down any “attempted” definitions of ${Dom}$ and ${Eq}$ that we like; but if our “attempted” definitions ever stop behaving well, then we (thereafter) “default” to something which is guaranteed to behave well. So, this does not limit our ability to define wand/set theories; it simply ensures that ${Dom}$ and ${Eq}$ will satisfy (dom1)–(dom3) and (eq1)–(eq3), even if our “attempt” misses the mark.

Here are the details. Let $D(w/a)$ be any formula which is loosely bound by ${\mathbb {L}{a}}$ , and let $E(w/a,u/b)$ be any formula which is loosely bound by ${\mathbb {L}{a}} \cup {\mathbb {L}{b}}$ . (Think of these as our “attempted” definitions of ${Dom}$ and ${Eq}$ .) Now we defineFootnote ³²

$$ \begin{align*} {Dom}(w/a) &\mathrel{\mathord{:}\mathord{\equiv}} Wand(w) \land D(w/a) \end{align*} $$

and

(1)

(2)

By (1) and (2): if D or E stops behaving well at some level, then ${Eq}$ collapses to strict identity at all higher levels. This all works as required to ensure that (dom1)–(eq3) hold (see Lemma 7.5). So our only constraint is this: we insist that our theory’s predicates ${Dom}$ and ${Eq}$ are defined in this way, from suitable predicates D and E.

5.5. Definition of a wand/set theory

I am now in a position to define the general notion of a wand/set theory. This makes precise the idea of a loosely constructive instance of the Wand/Set Template.

To recap: I have made some notational choices which impose a certain shape of formalism upon us; but these impose no substantial restrictions on our ability to flesh out the Template however we like (see, e.g., Section 2). My method for arranging entities into a hierarchy of wevels does nothing more than eliminate the need to talk about stages in our formalism (see Section 3). The Picture of Loose Constructivism yields a well-motivated explication of what it is to be “reasonably constructive” (see Section 4). The constraints on the notion of a wand’s “domain of action”, and of “equivalence” between wand-taps, are as permissive as they can be (see Section 5.4). I admit that I have gone slightly beyond the Template in making certain claims about the height of the hierarchy via WeakHeight, but these are extremely weak claims (see Section 5.2). Summing all this up: when we study all wand/set theories, in the sense of Definition 5.1, we study exactly the “reasonably constructive” ways to flesh out the Wand/Set Template (which accept a very weak principle about the hierarchy’s height).

At this point, readers may wish to check that all of the examples discussed in Section 1 can be rendered as wand/set theories. I leave Examples 1.1–1.4 to the reader, but exhaustively deal with Church’s [Reference Church and Henkin6] theory in Section 11.

7.1. Results about wevels

The following definition is helpful for reasoning about wevels:

Using this, we can replicate in S a sequence of elementary results which hold for LT (with tiny adjustments).Footnote ³⁵ I leave the details to the reader, but they culminate in two central results:

We can also show that the wevels behave nicely:

Arranging everything into well-ordered wevels has many nice consequences. For example, recall that wand-tapping is controlled by the defined predicates ${Dom}$ and ${Eq}$ (see Section 5.4); now simple reasoning about wevels shows that my method for ensuring the “good behaviour” of ${Dom}$ and ${Eq}$ works just as intended (I leave the proof to the reader):

Here is another important but simple consequence. If c is not bland, there is some rank-minimal a such that c is obtained by tapping a with some wand:

It will later be helpful to restate Lemma 7.6 in terms of ${Eq}$ rather than $\textit{Tap}$ :

This also yields an expected result: anything non-bland can be obtained from something bland using only finitely many wand taps:

Note that there is no uniqueness condition associated with Lemma 7.8; there may be many (equally short) paths to the same object.

7.2. Results about levels_u and hereditary-blandness

I now move from considering wevels to considering levels $_{\text {u}}$ . Unsurprisingly, given their similar definitions, wevels and levels $_{\text {u}}$ behave very similarly, provided we assume that $\text {u}$ has no self-membered elements.Footnote ³⁷ In detail, let CoreLev $_{\text {u}}$ be the theory whose axioms are Ext, all instances of Sep, and $(\forall x \in \text {u})\text {u} \notin \text {u}$ . Next, we use a new definition (cf. Definition 7.1):

The enthusiastic reader will now find it easy to prove these key facts:

These results allow us to talk about the $\alpha \text {th}$ level $_{\text {u}}$ , or ${V}^{\alpha }_{\text {u}}$ ,Footnote ³⁸ which immediately vindicates the recursive characterization of the levels $_{\text {u}}$ offered in Section 4.2:

In Section 4.2, I also “unofficially” defined the hereditarily bland sets as the denizens of $\textbf {V}_{\emptyset }$ , i.e., as members of any ${V}^{\alpha }_{\emptyset }$ , whilst “officially” offering Definition 4.3. To see that these definitions are equivalent, we just need to introduce the idea of the hereditarily bland part of a set; the rest is simple:

9.1. Defining the translation

It will help to start with an informal overview of how I will define the translation. I start by defining the things which will serve as the domain of interpretation; I call these things conches, as a rough portmanteau of “Conway” and “Church”. Here are basic ideas about conches.

• – We need a way record the result of “tapping” some earlier-found object, a, with some “wand”, $w$ . (Scare-quotes are needed, since this is all under interpretation.) We do this by considering the pair ${\langle w, a\rangle }$ .

• – We need to quotient such pairs under an equivalence relation (the interpretation of ${Eq}$ ); so a conch will typically be a set of such equivalent pairs. To ensure that we have a set of such pairs, we use a version of Scott’s trick, taking only the equivalent pairs of minimal rank (for some suitable notion of rank).

• – Finally, we need a way to consider “bland” entities; we take them to have the form $\{{\langle \emptyset , a\rangle }\}$ , which I write . (So I will insist that $\emptyset $ is not a “wand”, but a marker of “blandness”.)

Where $\sharp $ is the translation that I will define, I will set things up so that iff $x \in a$ . This allows me to compress most of the above into the following statement of intent (which I ultimately prove in Section 9.4):

Corollary 9.27. c is a conch iff either:

1. (1) , for some s whose members are all conches; or

2. (2) $c = \{{\langle w,a\rangle } \mathrel {:} \textit{Tap}_{\sharp }(w/a,c)\land {Mini}^{\sharp }(w/a)\} \neq \emptyset .$

Condition (1) will hold iff c is bland $_\sharp $ . Condition (2) will hold iff c is non-bland $_\sharp $ , whereupon ${\langle w,a\rangle } \in c$ iff c is found by “tapping” a with “wand” $w$ (and not by “tapping” anything of lower rank than a).

It should be clear that, if we are careful, we will indeed have interpreted S in $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ . It just remains to thrash through the details.

In S, the hereditarily bland sets form a sort of “spine” around which the rest of the hierarchy is constructed. So I will start by explaining how to simulate S’s hereditarily bland sets within $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ . In fact, this simulation is dictated by the choices I made above:

Note that $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ proves both that $\Theta $ is injective and that $\emptyset \neq \Theta x$ for any x (see Lemma 9.6). So, having used $\emptyset $ to code “bland” conches, I can say that a “wand” is any $\Theta $ -image of a $Wand$ . This motivates the following:

Definition 9.2. For any $w,x,y$ :

This definition forms the heart of my translation $\sharp : \mathsf{LT}_{\mathbf{\mathsf{S}}} \longrightarrow {\mathbf{\mathsf{S}}}$ . However, I cannot complete that translation until I have defined a suitable domain formula, i.e., until I have said what the conches will be. Unfortunately, this will takes some time.

In Section 4, I explicated the Picture of Loose Constructivism. The general idea was to treat the objects found at some stage of the hierarchy as urelements, and then to build a “bland-hierarchy” from those urelements. My first job is to translate this idea into $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ ; I do this by recursively defining $\textbf {U}_{\text {u}}$ (cf. Definition 4.1 and Lemma 7.12):

Definition 9.3 ( $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ )

For any set $\text {u}$ , recursively define (where this exists):

Say that $\textbf {U}_{\text {u}}(x)$ iff $\exists \alpha \ x \in U_{\text {u}}^{\alpha }$ .

The next task is to specify what the stages of our “ S-hierarchy” will look like. Specifically, I must define the conches and associate them with some notion of rank according, intuitively, to the stage at which they are first found.

To effect this, I define, for each ordinal $\sigma $ , the sets $\text {C}_{\sigma }$ , $\text {D}_{\sigma }$ , and $\text {E}_{\sigma }$ . Here are the guiding ideas:

• – $\text {C}_{\sigma }$ is a set whose members are the conches of rank $\leq \sigma $ (mnemonically: Conch).

• – $\text {B}_{\sigma }$ is a whose members are the conches of rank $< \sigma $ (mnemonically: Below).

• – I write $|a| = \sigma $ to indicate that a is a conch of rank $\sigma $ .

• – $\text {D}_{\sigma }$ and $\text {E}_{\sigma }$ respectively simulate ${Dom}$ and ${Eq}$ for conches of rank $\leq \sigma $ . Indeed, we will ultimately set things up so that:

  $$ \begin{align*} {\langle w, a\rangle} \in \text{D}_{\sigma}&\text{ iff}\ |a| \leq \sigma \text{ and}\ {Dom}^{\sharp}(w/a),\\ {\langle w,a,u,b\rangle} \in \text{E}_{\sigma}&\text{ iff}\ |a|,|b| \leq \sigma\ \text{ and}\ {Eq}^{\sharp}(w/a,u/b). \end{align*} $$

The guiding principles are quite clear; alas, the formal definition itself is quite long. Indeed, it is long enough for me to need to interrupt it with commentary to make it intelligible. Here is how it starts.

Definition 9.4 ( $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ )

For each ordinal $\sigma $ , let

where I write . Say that x is a conch, written ${Conch}(x)$ , iff $\exists \sigma \ x \in \text {C}_{\sigma }$ . For each conch x, let $|x|$ be the least $\sigma $ such that $x \in \text {C}_{\sigma }$ . tbc

This formal definition is incomplete (hence the “to be continued”). In particular, I have used the phrase “a $\tau $ -tap”, which I have not yet defined. However, I should pause to explain the definition for $\text {C}_{\sigma }$ . The component collects “bland” conches: when s is a set of conches, will act as a “bland” conch with exactly the members of s as “members”, and ’s rank will be the supremum of its “members”. Then $\{c \mathrel {:} (\exists \tau < \sigma )(c\text { is a }\tau \text {-tap})\}$ will collect the “non-bland” conches.

To complete Definition 9.4, then, I must define what I mean by a $\tau $ -tap; or, changing index for convenience, a $\sigma $ -tap. Roughly speaking, a $\sigma $ -tap should be a “non-bland” conch of rank $\sigma +1$ which was obtained by tapping some conch of rank $\sigma $ with some “wand”. More precisely :

Definition (9.4, cont.): For each ordinal $\sigma $ , say that c is a $\sigma $ -tap iff c is a non-empty set of ordered pairs such that, for all ${\langle w, a\rangle } \in c$ , these three conditions hold:

(tapr) $$\begin{align} &|a| = \sigma \text{ and }\text{if }{\langle w,a, u,b\rangle} \in \text{E}_{\sigma}\text{, then }|b| = \sigma. \end{align} $$

(tapd) $$\begin{align} &{\langle w, a\rangle} \in \text{D}_{\sigma}.\qquad\qquad\qquad\qquad\qquad\quad\ \kern2pt\qquad \end{align} $$

(tape) $$\begin{align} &{\langle w,a, u,b\rangle} \in \text{E}_{\sigma}\text{ iff }{\langle u, b\rangle} \in c.\qquad\qquad\ \kern1pt\qquad \end{align} $$

 tbc

Again, I will pause to explain this. Roughly: ( tapr ) says that a has rank $\sigma $ and that this rank is as small as possible; ( tapd ) says that a is within w’s “domain of action”; and ( tape ) says that c is an equivalence class of minimally ranked entities. But of course these clauses can only have this effect if we have appropriately defined $\text {D}_{\sigma }$ and $\text {E}_{\sigma }$ . So we offer :

Definition (9.4, cont.): For each ordinal $\sigma $ :

tbc

Interrupting again: whatever the exact meaning of the ${\tilde {\sigma }}$ -translation, it should be clear how the definitions of $\text {D}_{\sigma }$ and $\text {E}_{\sigma }$ are supposed to fit with our intentions (see the comments just before the start of Definition 9.4). So it just remains to define each ${\tilde {\sigma }}$ :

Definition (9.4, finished.): For each ordinal, $\sigma $ , define an identity-preserving translation ${\tilde {\sigma }}$ as follows (using $\Delta _I$ for I’s domain formula):

The domain formula, $\textbf {U}_{\text {C}_{\sigma }}$ , ensures that ${\tilde {\sigma }}$ quantifies only over entities which are “available” at the $\sigma \text {th}$ “wevel”. We then translate ${Bland}$ and $\in $ as suggested by Definition 9.2, whilst ensuring that we are dealing with entities in the intended “domain”. The clause for $Wand_{{\tilde {\sigma }}}$ is similar but more restrictive, and reflects the intention that “wands are hereditarily bland” (see Proposition 9.13 for more). Finally, the idea for $\textit{Tap}_{\tilde {\sigma }}$ is as follows: if c is found by “tapping” a with $w$ , then c is a $\tau $ -tap, for some $\tau < \sigma $ .

Definition 9.4 is finally complete. And I can use it to define the $\sharp $ -translation as the “limit” of the ${\tilde {\sigma }}$ -translations.

Definition 9.5 ( $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ )

Define $\sharp $ as an identity-preserving translation as follows:

I hope that my informal explanations make it plausible that $\sharp $ is, indeed, an interpretation. The remainder of this section simply verifies this fact.

9.2. Confirming the recursions

I must start by commenting on my use of recursive definitions. In brief: recursive definitions make good sense in $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ , but the defined term-function may be partial. Here is a slightly fuller explanation of this point.

Given a term, $\textbf {t}$ , we can stipulate (as usual) that f is an $\alpha $ -approximation to $\textbf {t}$ iff ${\text {dom}}(f) = \alpha $ and $(\forall \beta < \alpha )f(\beta ) = \textbf {t}(f\mathord {\restriction _{\beta }})$ . Ordinal induction will establish that $\alpha $ -approximations agree for any shared arguments, so we can write $f^{\textbf {t}}_\alpha $ for the unique $\alpha $ -approximation, if any exists. Where we then explicitly define

we can prove a version of the general recursion theorem:

$$ \begin{align*} \textbf{r}(\alpha) &= \textbf{t}(\{{\langle\beta, \textbf{r}(\beta)\rangle} \mathrel{:} \beta< \alpha\}),&&\text{if}\ f^{\textbf{t}}_{\alpha+1} \text{exists,}\\ \textbf{r}(\alpha)&\text{ is undefined,} &&\text{otherwise.} \end{align*} $$

All of this is as in ZF. The sole difference is that ZF proves that $f^{\textbf {t}}_{\alpha +1}$ must exist for any $\alpha $ , and $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ might not prove this. So $\textbf {r}$ may be partial.

It is, then, important to confirm that many of the notions defined in Section 9.1 are total. To do this, we can take a slight shortcut. Pseudo-WeakHeight ensures that there is no last level. So, where $\textbf {r}$ is recursively defined from $\textbf {t}$ as above, to show that every $\textbf {r}(\alpha )$ exists, we need only show that there must always be an ordinal corresponding to “what $\textbf {r}(\alpha )$ ’s rank would have to be”.

We will first use this strategy to show that $U_{\emptyset }^{\alpha }$ and $\Theta $ are total.

Proof Where $\alpha $ is any ordinal, write it as $\alpha = \lambda _\alpha + n_\alpha $ , with $\lambda _\alpha $ either a limit ordinal or $0$ , and $n_\alpha $ a natural number. By induction, $\text {R}(U_{\emptyset }^{\alpha }) = \lambda _\alpha + 4n_\alpha $ , so each $U_{\emptyset }^{\alpha }$ exists.

A simple $\in $ -induction shows that $\Theta $ is injective. Suppose for induction that $b \in U_{\emptyset }^{\text {R}b+1}$ whenever $b \in a$ . As $\text {R}a = \operatorname{lsub}_{b\in a}{\text {R}b}$ , we have , so $\Theta a \in U_{\emptyset }^{\text {R}a+1}$ . Suppose also that $\text {R}b = |\Theta b|$ whenever $b \in a$ ; then

It follows that $\Theta : V \longrightarrow \textbf {U}_{\emptyset }$ is total. For surjectivity, suppose for induction that if $x \in U_{\emptyset }^{\gamma }$ then $\Theta ^{-1}x$ exists, whenever $\gamma < \beta $ . Let $b \in U_{\emptyset }^{\beta }$ ; then exists by Separation, and and .

Note that $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ might not prove that $U_{\text {u}}^{\alpha }$ exists for every $\alpha $ and every $\text {u}$ . This is not a problem; we do not in general require that it does. (Similarly: S might not prove that ${V}^{\alpha }_{\text {u}}$ exists for every $\text {u}$ and $\alpha $ .) Unsurprisingly, though, we do require that $\text {C}_{\alpha }$ exists for every $\alpha $ . And it provably does. Indeed (Pseudo-)WeakHeight was chosen precisely to secure this fact.

Proof It suffices to establish the result for $\text {C}_{\alpha }$ . Let $\Omega = \operatorname{lub}_{w\text { a wand}}\text {R}(\Theta w)$ . Writing each $\alpha $ as $\lambda _\alpha + n_\alpha $ , I claim that

$$ \begin{align*}\text{R}\text{C}_{\alpha} \leq \Omega + \lambda_\alpha + 4n_\alpha + 4.\end{align*} $$

By Lemma 9.6, there is m such that $\Omega = \text {R}(\{w \mathrel {:} Wand(w)\}) + m$ ; so $\text {C}_{\alpha }$ ’s required rank exists by my claim and Pseudo-WeakHeight.

It just remains to establish my claim. This is by induction. Trivially, $\text {R}\text {C}_{0} = 4$ .

Successor case. Suppose $\text {R}\text {C}_{\alpha } \leq \Omega + \lambda _\alpha + 4n_\alpha + 4$ . If $s \subseteq \text {B}_{\alpha +1}=\text {C}_{\alpha }$ , then

If c is a $\beta $ -tap with $\beta \leq \alpha $ , then c’s members are of the form ${\langle \Theta w, d\rangle }$ , with w a wand and $\text {R}d \leq \Omega + \lambda _\alpha + 4n_\alpha + 3$ , and since $\text {R}({\langle \Theta w, d\rangle }) = \max (\text {R}\Theta w, \text {R}d) + 2$ , we obtain:

$$ \begin{align*} \text{R}c &\leq \Omega + \lambda_\alpha +4n_\alpha+6. \end{align*} $$

So $\text {R}\text {C}_{\alpha } \leq \Omega + \lambda _\alpha + 4n_\alpha + 8$ , as required.

Limit case. Using the induction hypothesis, if $s \subseteq \text {B}_{\alpha }$ then . So $\text {R}(\text {C}_{\alpha }) \leq \Omega + \alpha + 4$ , as required.

9.3. $\star $ -translations and blandness $_\star $

I will now look at the $\sharp $ and ${\tilde {\sigma }}$ translations. For readability, I use $\star $ to stand ambiguously for either $\sharp $ or any ${\tilde {\sigma }}$ . So if I assert that , I mean that and for any $\sigma $ . Recall that $\Delta _\star $ is always $\star $ ’s domain-formula.

I start with some elementary results about bland $_\star $ conches. These results are pretty immediate consequences of the definitions (and previous results) and I leave their proofs to the reader:

I now want to consider hereditarily bland $^\star $ conches. In particular, I want to show that $\Theta $ is an “isomorphism” to the hereditarily bland $^\star $ conches:

It follows that hereditary blandness $^\star $ does not depend on the choice of $\star $ . We also obtain the $\star $ -translations of any S-axioms which only concern hereditarily bland sets:

9.4. Wevels $^\star $ and levels $_{\text {u}}^\star $

I will now move on to considering “wand taps”, starting with some easy facts:

This allows us to describe the moment when a conch is “first found”:

The next challenge is characterize the wevels $^\star $ . This is easier than we might fear. The basic facts about wevels follow from CoreWev (see Sections 3 and 7.1); so the $\star $ -translations of those basic facts hold by Lemmas 9.8 and 9.12. Specifically: $\in _\star $ well-orders the wevels $^\star $ (by Theorem 7.3 $^\star $ ), and a wevel $^\star $ is exactly the bland $_\star $ set of everything found-earlier $^\star $ (by Lemma 7.2 $^\star $ ). Once we combine this with Lemma 9.16, we can characterize the wevels $^\star $ in terms of the $\text {B}_{\alpha }$ ’s:

Proof Whenever $\Delta _\star (c)$ , Lemma 9.16 tells us that $|c| < \alpha $ iff . So suppose for induction that is the $\beta \text {th}$ wevel $^\star $ , whenever $\beta < \alpha $ . Then by Lemma 9.11, the earlier biconditional becomes

So is the $\alpha \text {th}$ wevel $^\star $ , by Lemma 7.2 $^\star $ . Now use induction, Theorem 7.3 $^\star $ .

Combining Lemmas 9.16 and 9.17 allows us to characterize ${\mathbb {L}^\star }$ :

This delivers the $\sharp $ -translation of two of S’s axioms:

Note that this only gives us the $\sharp $ -translations of Strat and WeakHeight. Indeed, we should not expect their ${\tilde {\sigma }}$ -translations to hold: denizens of $\Delta _{{\tilde {\sigma }}}$ must sit at some level but may come after any wevel $^{\tilde {\sigma }}$ . Our next job, then, must be to characterize the levels. This can be done along the same line as the wevels $^\star $ , mimicking the proof of Lemma 9.17:

Now that we understand wevels $^\star $ and levels, we can $\star $ -translate the central notion used in the Picture of Loose Constructivism, i.e., $\textbf {V}_{\text {u}}$ :

Lemma 9.21 ( $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ )

If , these are equivalent predicates:

This delivers a key fact: satisfaction of loosely bound formulas is preserved as we expand our interpretations, moving (with $\sigma < \tau $ ) from ${\tilde {\sigma }}$ to ${\tilde {\tau }}$ and ultimately to $\sharp $ :

Proposition 9.22 ( $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ , scheme)

Let be loosely bound by $\max ({\mathbb {L}{v_1}}, \ldots , {\mathbb {L}{v_n}})$ . When $\max (|c_1|, \ldots , |c_n|) \leq \sigma < \tau $ , these are equivalent:

Corollary 9.23 ( $\mathsf{LT}_{\mathbf{\mathsf{S}}}$ )

Lemma 7.5 $^\sharp $ holds, so that, e.g., ${Eq}^\sharp $ is an equivalence relation. Furthermore, these formulas are equivalent (row-wise), when $|a|, |b| \leq \sigma $ :

$$ \begin{align*} {Dom}^{\sharp}(w/a)&& {Dom}^{{\tilde{\sigma}}}(w/a) && {\langle w,a\rangle} \in \mathrm{D}_{|a|}\\ {Eq}^{\sharp}(w/a,u/b)&& {Eq}^{{\tilde{\sigma}}}(w/a,u/b) && {\langle w,a,u,b\rangle} \in \mathrm{E}_{\max(|a|{, |b|}).} \end{align*} $$

Using Corollary 9.23 freely and frequently from this point onwards, we are at last in a position to show that $\sharp $ is an interpretation:

Assembling all the $\sharp $ -translations of S’s axioms, we have what we required:

We can also obtain the Corollary which I stated as a target in Section 9.1, via an easy lemma:

Proof Suppose $\textit{Tap}_{\sharp }(w/a,c)$ and let

$$ \begin{align*} e &= \{{\langle u,b\rangle} \mathrel{:} {Eq}^{\sharp}(w/a,u/b) \land {Mini}^{\sharp}(u/b)\}. \end{align*} $$

To see $c \subseteq e$ , fix ${\langle u,b\rangle } \in c$ . Now $\textit{Tap}_{\sharp }(u/b,c)$ by Lemma 9.15(3), and so ${Eq}^{\sharp }(w/a,u/b)$ by Equating $^\sharp $ . Also, if ${Eq}^{\sharp }(u/b,v/c)$ , then $|b| \leq |c|$ by ( tapr ), i.e., $\text {R}^{\sharp }b \leq ^\sharp \text {R}^{\sharp }c$ by Lemma 9.18, so ${Mini}^{\sharp }(u/b)$ . To see $e \subseteq c$ , keep ${\langle u,b\rangle } \in c \neq \emptyset $ as before, and suppose ${\langle u', b'\rangle } \in e$ . Now ${Eq}^{\sharp }(u/b,u'/b')$ , and since ${Mini}^{\sharp }(u'/b')$ we have ${\langle u',b'\rangle } \in c$ by ( tape ).

11.1. Axiomatizing Church’s [Reference Church and Henkin6] theory as CUS

In this sub-section, I define CUS, my version of Church’s Universal Set theory. The basic idea is to help myself to the notion of a wevel, and then write down the axioms that would be obviously suggested by the discussion in Section 1.3.

CUS’s signature is the same as any wand/set theory. It has the quasi-notational axioms InNB, TapNB, and TapFun. It also has the axioms which arrange everything into well-ordered wevels—Ext, Sep, and Strat—and the axiom WeakHeight.

As indicated in Section 1.3, I want to treat $0$ as the complement wand, and each $n> 0$ as the cardinal $_n$ wand. So Church’s wands will be the natural numbers:

• Wands _c $\{w \mathrel {:} Wand(w)\} = \omega .$

Note that this entails HebWands, and it is formulated solely in terms of hereditarily bland sets and $Wand$ (see Section 5.3). So far, then, CUS is on track to be a wand/set theory.

I now need to provide axioms which specify how the wands behave. To explain the $\mathsf{cardinal}_n$ wands, I need to introduce Church’s notion of n-equivalence (though see Caveat 5 of Section 11.2):Footnote ⁴³

Note that $a \approx _1 b$ iff a and b are (non-empty, bland) sets which are equinumerous in the usual sense.

I now need axioms to say exactly when should exist, for any wand n and any object a. We want to say: tapping anything with complement yields an object; tapping anything bland with any cardinal $_n$ yields an object (if the n-equivalence class would be non-empty); and tapping anything non-bland with any cardinal $_n$ yields nothing. But there is a wrinkle. Suppose b is bland; since we are treating $0$ as complement, we expect that ;Footnote ⁴⁶ but this would violate TapNB.

To retain TapNB, we will say that we find an object by tapping a with complement iff a is not itself the complement of anything bland. (As foreshadowed in Section 2, I will eventually define a more expansive notion, according to which tapping with $0$ will yield b itself; see Definition 11.2.) We effect this by stipulating:

• Making _c

Note that this does not have the same shape as Making, so we have departed from the shape of axioms stated in Definition 5.1. (I return to this in Section 11.4.)

Our next axioms govern the complement wand:

• Comp1 if , then $a = b.$

• Comp2 if , then

These say that is injective, and that double- is identity (provided this does not require taking for any bland b). Last, we have axioms governing the cardinal $_n$ wands:

• if $0 < m < \omega $ and $0 < n < \omega $ , then:

• Card1 ,

• Card2 , and

• Card3

So a and b have the same cardinality $_n$ iff a and b are n-equivalent; if $n \neq m$ then no cardinal $_n$ is a cardinal $_m$ ; and no cardinal $_n$ is the complement of anything bland or of any cardinal $_m$ .

We now define CUS as the theory whose axioms are: InNB, TapNB, TapFun, Ext, Sep, Strat, WeakHeight, Wands _c , Making _c , Comp1–Comp2, and Card1–Card3.

11.2. Caveats and more expansive notions

With CUS, we have a cleanly presented version of Church’s [Reference Church and Henkin6] theory. Or so I claim; I should now offer some small caveats.

Caveat 1: Church had ZF in mind. More specifically, Church wanted (what I call) the hereditarily bland sets to obey ZF. This causes no concern: if we want to follow Church here, we can augment CUS with the axiom scheme of Replacement, restricted to hereditarily bland sets. This will not affect our synonymy result; rather, the resulting theory will be synonymous with ZF itself.

Caveat 2: Church did not speak about wands. In a sense this is immaterial; talking about wands is only a helpful heuristic. But there is an important formal point: CUS’s signature has primitives ${Bland}$ , $\in $ , $Wand$ and $\textit{Tap}$ ; Church’s own theory has just one primitive, $\in $ . Fortunately, this is incidental: in Section 11.3, I show that we can reformulate CUS using just $\in $ .

Caveat 3: I have a restrictive notion of tapping. As explained: if b is bland, then CUS proves that does not exist, but we intuitively want tapping b twice with $0$ to yield b. To address this, I simply introduce a more expansive notion of tapping (whose good behaviour is confirmed in Lemma 11.7):

Caveat 4: I have a restrictive notion of membership. I have been glossing as “a’s complement”. However, I have said nothing, yet, which indicates that ’s members should be exactly those x such that $x \notin a$ . Worse: whilst should be “the universal set”, CUS’s axioms TapNB and InNB entail that . Evidently, I need to define a more expansive notion of membership; here it is:

Definition 11.3 (CUS)

When c is bland and $0 < n < \omega $ , define:

We confirm this covers all cases in Lemma 11.6.

Caveat 5: Church’s definition of $\approx _n$ is not quite my own. Relatedly: Church defined $\approx _n$ using an “expansive” notion of membership, according to which everything is a member of the (non-bland) universal set. I defined $\approx _n$ using the “restrictive” notion of membership, $\in $ , according to which only bland sets have members. (I am forced to do things this way, because I invoke $\approx _n$ when I define the “expansive” notion of membership, $\mathrel {\varepsilon }$ , in Definition 11.3.)

Again, the difference is unimportant. Church’s definition of $\approx _j$ only shows up in his axiomatic theory in the following axiom:Footnote ⁴⁷

• L_j : if a is well founded, then $\{x \mathrel {:} a \approx _j x\}$ is a set.

Now suppose that a is well-founded and that $a \approx _j x$ . Then x’s members are bland,Footnote ⁴⁸ and so are the members of members, and so on, digging j-levels deep through membership chains. So, as it occurs in $\text {L}_j$ , either definition would have the same effect.

However, there is a more subtle difference. Church’s principle $\text {L}_j$ is not a single axiom; instead, he has one axiom for each natural number $j> 0$ , with numbers supplied in the metalanguage. This suggests that he has defined j-equivalence by metalinguistic recursion rather than object-linguistic recursion. By contrast, my Definition 11.1 is fully object-linguistic, and the success of Definition 11.3 requires this fact.

I conjecture that Church relied on metalinguistic recursion because he had not proved a suitable object-linguistic recursion theorem.Footnote ⁴⁹ Fortunately, I have such a theorem (cf. Section 9.2), and I am not afraid to use it.

Caveat 6: Church had no Beschränktheitsaxiom. Footnote ⁵⁰ The axiom Strat acts as an axiom of restriction, arranging everything into a well-founded hierarchy. Indeed, CUS proves Lemma 7.8, which states that anything non-bland is obtained by starting with something bland and tapping it finitely many times with some wands (i.e., some object-linguistic natural numbers). By contrast, Church’s theory is open-ended: for all we know, its universe contains objects obtained by using hitherto-unmentioned wands, or beasts which float around outside of any hierarchy.

Deleting Strat from CUS would certainly destroy my proof-strategy for obtaining synonymy. (Indeed, I suspect it would actually surrender synonymy.) So: including Strat genuinely changes Church’s theory.

That said: this change to the theory does not not flag a departure from Church’s approach. Church established the consistency of his [Reference Church and Henkin6] theory by outlining what is now called a “Church–Oswald model”,Footnote ⁵¹ and the denizens of Church–Oswald models are always arranged into a nicely well-founded hierarchy. So the model theory of Church-style approaches has always implicitly invoked a Beschränktheitsprinzip, even if his formal theory lacked an explicit Beschränktheitsaxiom. Indeed: my framework of wand/set theories really just provides a framework for turning a description of a Church–Oswald model into a formal theory with a Beschränktheitsaxiom, which is then (by my Main Theorem) synonymous with a ZF-like theory.

11.3. CUS as a theory of nothing but sets

I now want to show that CUS does what we want it to do. I begin with two trivial results concerning , whose proof I leave to the reader:

The next result says that, in CUS, everything is (exclusively) either: bland, the complement of something bland, some cardinality $_n$ of something bland, or the complement of some such cardinality $_n$ .

This result vindicates my expansive notion of “membership”. Roughly, we want to say that x is in a’s cardinal $_n$ iff x and a are n-equivalent; and that x is in a’s complement iff x is not in a; we stipulate this with Definition 11.3, and can see by Lemma 11.6 that this exhausts all cases.

It is obvious from Definition 11.3 that each $n> 0$ behaves as the cardinal $_n$ wand. It is only slightly less obvious that $0$ behaves as the complement wand. Specifically, using Definition 11.2 and writing for $\lnot (x \mathrel {\varepsilon } y)$ :

This yields a generalized version of extensionality:

Since the objects of CUS obey this generalized extensionality principle, we can treat CUS as a theory of nothing but sets. Indeed, as claimed in Caveat 2 of Section 11.2, we can use Lemmas 11.6–11.8 to reformulate CUS using only a membership-predicate.

Here is one way to do this. Start by defining an identity-preserving translation with these atomic clauses (and an unrestricted domain-formula):

The translation is well-defined, since $\approx _n$ is defined without using the primitive $\textit{Tap}$ . Now $\in $ is the only primitive of $\mathsf{CUS}^\bullet $ , i.e., the $\bullet $ -translation of CUS.

Trivially, $\bullet $ is an interpretation $\mathsf{CUS}\longrightarrow \mathsf{CUS}^\bullet $ . We can also define a translation, $\circ $ , in the opposite direction, via $x \in ^\circ a \mathrel {\mathord {:}\mathord {\equiv }} x \mathrel {\varepsilon } a$ . By running these translations back-to-back, it is easy to show that CUS and $\mathsf{CUS}^\bullet $ are synonymous. Hence CUS can be reformulated, without gain or loss, as a theory whose only primitive is $\in $ , and which obeys Extensionality.

I leave the proof of this claim to keen readers, but here are the main steps. Using Lemmas 11.6 and 11.8, show that CUS proves: ${Bland}(a) \leftrightarrow {Bland}^{\bullet \circ }(a)$ ; and $x \in a \leftrightarrow x \in ^{\bullet \circ } a$ ; and $Wand(w) = Wand^{\bullet \circ }(w)$ ; and $\textit{Tap}(w/a,c) \leftrightarrow \textit{Tap}^{\bullet \circ }(w/a,c)$ . Then use Lemmas 11.6 $^\bullet $ and 11.8 $^\bullet $ to show that $\mathsf{CUS}^\bullet $ proves $x \in a \leftrightarrow x \in ^{\circ \bullet } a$ . This establishes synonymy, and $\mathsf{CUS}^\bullet $ proves Extensionality by Lemma 11.8 $^\bullet $ .

11.4. CUS is a wand/set theory

We have seen that CUS does everything we wanted it to. But my last task is to show that CUS is a wand/set theory, in the sense of Definition 5.1. To this end, I will close by presenting a wand/set theory, WSC, for Wand/Set Church theory, and showing that it is just an alternative axiomatization of CUS.

WSC’s first few axioms are: InNB, TapNB, TapFun, Ext, Sep, Strat, WeakHeight, and Wands _c . These are shared with CUS. WSC’s remaining axioms, Making and Equating, arise by explicitly defining two formulas, D and E (see Section 5.4). To define D, I simply insert a loose bound on the formula in the right-hand-side of Making _c :Footnote ⁵²

To define E, I simply list the few cases under which we would want to identify wand-taps, whilst inserting some loose bounds when required:

$$\begin{align*}\hspace{-13em}E(m/a,n/b) &\mathrel{\mathord{:}\mathord{\equiv}} m, n \in \omega\text{ and:} \end{align*}$$

(e1) $$\begin{align} & (m = n \land a = b) \text{ or} \end{align} $$

(e2) $$\begin{align} &\hspace{3em} \big(0 < m = n \land a \approx_m b\big) \text{ or} \end{align}$$

(e3)

(e4)

Note: From here onwards, I reserve D and E for these defined predicates, and I reserve ${Dom}$ and ${Eq}$ for the predicates we obtain from D and E as in Section 5.4.

It just remains to show that $\mathsf{WSC} = \mathsf{CUS}$ . The crucial insight is that, because everything is arranged neatly into wevels, the explicitly stipulated loose bounds have no real effect. Concerning D: if x is bland and , then we “know” that x is found before a. Concerning E: if there is some $d \approx _n b$ with , then there is some $e \approx _n d$ of minimal rank; so and we “know” that both e and are found before a.

It just remains to prove what I have claimed to “know”. My approach is to build up a reservoir of principles shared by WSC and CUS, until the theories come to coincide.