2.1 H-extensionality and metric set structures

The structures we will be considering will be of the form , where $(M,d)$ is a complete metric space, and is a closed binary relation. As is suggested by the notation, is meant to be interpreted as a set membership relation, and, as such, we would like for it to be extensional. Obviously we could just require extensionality of as a binary relation in the standard sense, but for a few different reasons, we will opt to place a stronger condition on . To define this condition, recall one of many equivalent definitions of the Hausdorff distance between subsets of a metric space.

On the full power set of a metric space M, $d_{\mathrm {H}}$ is an extended pseudo-metric, but on the collection of close subsets of M, it is an extended metric. We will be concerned exclusively with $[0,1]$ -valued metrics. In this context, it makes sense to modify the above definition to take $d_{\mathrm {H}}$ to be $[0,1]$ -valued as well. This only changes the distance between the empty set and non-empty sets. In particular, $d_{\mathrm {H}}(\varnothing , A) = 1$ for any non-empty A. This is the definition of the Hausdorff distance we will actually use.

The form of Definition 2.1 above makes it clear that $d_{\mathrm {H}}$ is a direct metric generalization of extensional equality of sets. $A=B$ if and only if for every $a \in A$ , there is a $b \in B$ such that $a=b$ and for every $b \in B$ , there is an $a \in A$ such that $a=b$ . In this way, we take as our extensionality axiom a direct translation of the statement ‘ $A = B$ if and only if A and B are coextensive.’

Metric set structures are a generalization of extensional digraphs (i.e., discrete models of the extensionality axiom). If $\delta $ is a $\{0,1\}$ -valued metric on V, then $(V,\delta ,E)$ is a metric set structure if and only if $(V,E)$ is an extensional digraph.

We should note that the definition of $\mathrm {H}$ -extensionality contains a somewhat arbitrary choice. After all, we could have just as easily required that . That said, the choice we have made is reasonable and seems to work well, so we have not investigated other possibilities in this article. Also, we should note that this notion of extensionality appears in a previous paper of the author [Reference Hanson9, Definition 6.4].

A commonly cited benefit of extensionality is that it allows one to take $\in $ as the only primitive notion, with $x=y$ being defined as . It seems unlikely that we will be able to do something similar with , but we can do something similar with the natural $[0,1]$ -valued version of , which is the distance from x to the elements of Y, commonly written $\operatorname {\mathrm {dist}}(x,Y)$ or $d(x,Y)$ . In the context of a set theory, $d(x,Y)$ is entirely unacceptable, being tantamount to writing $x = Y$ to mean $x \in Y$ . $\operatorname {\mathrm {dist}}(x,Y)$ is too long to use frequently, so we will use the following notation (which is used similarly in [Reference Chang3, Reference Fenstad4]).

Another useful characterization of the Hausdorff metric is this: $d_{\mathrm {H}}(A,B) = \sup _z|\operatorname {\mathrm {dist}}(z,A)-\operatorname {\mathrm {dist}}(z,B)|$ . This means that if is a metric set structure, we have that $d(x,y) = \sup _z|e(z,x)-e(z,y)|$ . Since if and only if $e(x,y) = 0$ , it should be possible to take $e(x,y)$ as our only primitive notion. This is essentially the approach taken in the formalization of (see [Reference Hájek8] and [Reference Montagna17, Chapter 4.5]) and this is the approach we will take in the ‘official’ continuous logic formulation of $\mathsf {MSE}$ , which we will discuss in Section 6.

2.2 Formulas

Our formalism will be a small modification of first-order continuous logic, introduced in its modern form in [Reference Yaacov, Alexander Berenstein, Henson and Usvyatsov25]. Our only predicate symbol will be the metric, $d(x,y)$ , but we will take bound quantifiers of the form as a primitive notion. Note though that we cannot access as a formula directly.

Definition 2.4. Our set of formulas, written , is the smallest non-empty set of expressions satisfying the following: For any , variables x and y, and $r \in \mathbb {R}$ , contains the expressions:

When we need to be more specific, we will refer to elements of as -formulas.

We refer to quantifiers of the form or as bounded. The free variables of a formula $\varphi $ are defined in the obvious way. We write $\varphi (\bar {x})$ to indicate that the free variables of $\varphi $ are included in $\bar {x}$ .

For bookkeeping purposes, we will need the inductively defined quantity given by

• • $v(1) = v(d(x,y)) = 1$ ,

• • $v(\varphi + \psi )=v(\varphi )+v(\psi )$ ,

• • $v(\max (\varphi ,\psi ))=v(\min (\varphi ,\psi ))=\max (v(\varphi ),v(\psi ))$ ,

• • $v(r \cdot \varphi ) = |r|v(\varphi )$ ,

• • .

We are allowing ourselves arbitrary real numbers in Definition 2.4 because it will be convenient in several places. This convenience comes at a cost later in Section 6, however.

The intended interpretation of a given formula is clear, although we do need to specify the behavior of bounded quantifiers over empty sets. This is the first of two reasons why we defined the quantity $v(\varphi )$ .

The conventions regarding suprema and infima of empty sets were chosen so that formulas would always be real-valued (rather than taking on values in $\mathbb {R}\cup \{\pm \infty \}$ ) and so that $\sup $ and $\inf $ are monotonic with regards to set inclusion, although we have to prove that this is actually the case.

We will often use commonsensical shorthand, such as $\varphi +\psi +\theta $ for $\varphi +(\psi +\theta )$ , $\varphi - \psi $ for $\varphi + (-1)\cdot \psi $ , and $|\varphi |$ for $\max (\varphi ,-\varphi )$ . We will abbreviate consecutive quantifiers with expressions such as $\sup _{xy}$ . By an abuse of notation, we will also write $e(x,y)$ as shorthand for the formula .

In the context of a metric set structure M, we may often refer to formulas with parameters, such as $\varphi (\bar {x},\bar {a})$ for some $\bar {a} \in M$ , as formulas and write them with parameters suppressed.

Two important properties of formulas in continuous logic are that they only take on values in some bounded interval and that they are always uniformly continuous. The relevant interval and modulus of uniform continuity can be determined by the formula alone. We will need similar facts here.

2.3 Excision

Our comprehension scheme is better defined in terms of its important consequence, rather than directly, as it takes a proof to establish that this property is even axiomatizable. The principle can be informally justified like this:

Suppose that we run a chalk factory and we are contractually obligated to produce pieces of chalk that are no longer than 7.62 cm in length. (The chalk is boxed by another company and needs to fit in their boxes.) Of course, our machine, being cheap, actually produces pieces that are anywhere between roughly 7.4 cm and 7.8 cm. To deal with this, we add a second machine that measures length and rejects pieces that are too long. To maximize our output, we might say that we want it to reject a piece if and only if its length is strictly longer than 7.62 cm, but the realities of physical measurement mean that this is impossible to actually accomplish. Since the penalties for violating the contract are quite harsh, we need to give ourselves some leeway, but we also want to make sure we aren’t throwing away too many acceptable pieces of chalk. So we configure the machine to accept chalk if it measures it to be no longer than 7.6 cm. We know that the error of the machine is no more than 0.01 cm, so we can guarantee that we will accept any piece of length at most 7.58 cm and reject any piece of length 7.62 cm or more, but we do not make any promises about the behavior of the machine in the gap between these bounds.Footnote ⁵

This is the manner in which we will approximate comprehension. Given a formula $\varphi (x,\bar {y})$ (i.e., a ‘measurable quantity’), bounds $r < s$ , and parameters $\bar {a}$ , we promise that we can deliver a set b such that for any c, if , then , and if $\varphi (c,\bar {a}) \geq s$ , then , but we make no commitment about those c’s for which $r < \varphi (c,\bar {a}) < s$ .

It is straightforward but worthwhile to see how this principle avoid Russell’s paradox. We can consider a set $a_r$ satisfying that if , then and if , then $1-e(b,b) < 1$ . As we pick r closer and closer to $1$ , we get better and better approximations of the Russell class, but for each r, we consistently have that $0 < e(a_r,a_r) < 1-r$ . So we see that while our theory is strictly speaking a $[0,1]$ -valued set theory like , there is something of a qualitative difference in its avoidance of Russell’s paradox. While is possiblyFootnote ⁶ able to avoid Russell’s paradox by Brouwer’s fixed point theorem, our theory avoids it by virtue of the required gap between r and s (although these are not unrelated phenomena).

We are now finally able to define the class of models of our theory directly before defining the theory itself.

The following notation will be useful.

Definition 2.9. Given a metric set structure M, a formula $\varphi (x,\bar {y})$ , tuple $\bar {a} \in M$ , and reals $r < s$ , we write

to mean that for any c, if , then and if , then $\varphi ^M(c,\bar {a}) < s$ .

Note of course that is not a uniquely specified object, but if M satisfies excision, it always exists.

It is immediate to show that models of $\mathsf {MSE}$ contain some of the familiar sets one expects to see in a set theory with a universal set.

Since these sets are unique by $\mathrm {H}$ -extensionality, we will write $\varnothing ^M$ for and $V^M$ for . We may drop the superscript M if no confusion will arise.

3.1 Relative excision

An important common construction in set theory is separation, i.e., comprehension relative to a given set. Ordinarily, separation is an easy consequence of comprehension— $\{x \in A: \varphi (x)\}$ is the same as the set $\{x : x \in A \wedge \varphi (x)\}$ —but seeing that excision is merely an approximate form of comprehension, one might worry that we will only be able to find sets that are approximately subsets of other given sets. In other words, if B is a rough approximation of $\{x : x \in A \wedge \varphi (x)\}$ , then it would only be the case that for some small but positive r. Fortunately, we are able to build exact subsets of a given set and thereby perform relative excision.

The following is a special case of relative excision, but we state it first because it is the only form of relative excision we will actually use and it is much easier to prove.

Lemma 3.3. Fix $M \models \mathsf {MSE}$ . For any $a,b \in M$ and $\varepsilon> 0$ , there is a $c \sqsubseteq a$ such that

Proof. Fix $\varepsilon> 0$ . Let $c_0 = b$ . For any n, let . At stage n, given $c_n$ , let

Note that for any $f \in M$ , if and $e(f,c_n) < t_n + 2^{-n-4}\varepsilon $ , then , and if , then $e(f,a) < 2^{-n-4}\varepsilon $ and $e(f,c_n) < t_n + 2^{-n-3}\varepsilon $ . In particular, . Note also that by the definition of $t_n$ , we have that for any , there is an such that $d(f,g) < t_n + 2^{-n-4}\varepsilon $ . Such a g must be an element of $c_{n+1}$ . Since we can do this for any , we have that . Hence, for any $n> 0$ , we have that . Therefore $(c_n)_{n\in \mathbb {N}}$ is a Cauchy sequence. Let .

Since $t_n \to 0$ as $n \to \infty $ , we have that $c \sqsubseteq a$ . Now we just need to verify that . Our estimates give that

as required.

In light of Propositions 3.2 and 3.4, we will write

to mean that for any $f \in M$ , if and , then and if , then and $\varphi ^M(f,\bar {a}) < s$ .

3.2 Comprehension for definable classes

Given Proposition 3.4, one might be tempted to ask whether we can just outright show that models of $\mathsf {MSE}$ satisfy a more conventional form of comprehension. Suppose we have a formula $\varphi (x)$ and we wish to form the set $\{x \in M : \varphi ^M(x) = 0\}$ . Could we not just form the sequence and take the limit? While we are perfectly able to form this sequence externally, the difficulty is that it will in general fail to be a Cauchy sequence.

Regardless, there are times when such a sequence of approximations does actually converge in the Hausdorff metric, giving us an instance of exact comprehension. This happens precisely when $\{x \in M: \varphi ^M(x) = 0\}$ is a definable set in the sense of continuous logic, although in the context of a set theory it would be more appropriate to refer to these as definable classes.

This definition is perhaps most strongly motivated by the fact that definable classes are precisely those that admit relative quantification.

Lemma 3.6. For any metric set structure M and $\bar {a} \in M$ , if $\varphi (\bar {x},\bar {a})$ is the distance predicate of a definable class $D \subseteq M^n$ , then for any $\psi (\bar {x},\bar {y},\bar {z})$ and $\bar {c} \in M$ ,

where $\inf \varnothing $ is understood to be $v(\psi )$ .

In continuous logic generally, definable classes can be characterized as those sets that admit relative quantification in the same sense as Lemma 3.6. In models of $\mathsf {MSE}$ moreover, definable classes of $1$ -tuples correspond precisely to sets. In particular, every definable class is explicitly definable by $e(x,a)$ for some a.

Proposition 3.7 allows us to answer some very basic questions that we haven’t resolved yet.

Unfortunately, however, it is generally not the case that the distance to the intersection of two sets X and Y can be computed from the distances to X and Y. As such, we cannot establish the existence of intersections in general, and indeed as demonstrated in Proposition B.1 such intersections do not always exist. We do, however, get a kind of approximate intersection in the form of .

A minor corollary of Corollary 3.8 is that models of $\mathsf {MSE}$ satisfy $d(x,y) = \sup _{z}|e(x,z)-e(y,z)|$ , as witnessed by the singleton $\{x\}$ . In $[0,1]$ -valued set theories, the quantity $1-\sup _z|e(x,z)-e(y,z)|$ is often referred to a Leibniz equality, as it represents the degree to which x and y cannot be discerned from each other.

In light of Corollary 3.8, we will write $\{a_0,a_1,\dots ,a_{n-1}\}$ for the finite set containing $a_0,a_1,\dots ,a_{n-1}$ , $a \sqcup b$ for the union of a and b, and $\overline {\bigsqcup a}$ for the closure of the union of the elements of a.

Now that we have the ability to form finite sets by Corollary 3.8, we are free to code ordered pairs. While we certainly could use the standard Kuratowski ordered pair, Wiener’s earlier definition is actually preferable to us for technical reasons.Footnote ⁸ As such, we will write for $\{\{\{a\},\varnothing \},\{\{b\}\}\}$ . Recall that .

Lemma 3.9. Let $M\models \mathsf {MSE}$ . For any $a,b,c,f \in M$ ,

Proof. Let $A = \{\{a\},\varnothing \}$ , $B = \{\{b\}\}$ , $C = \{\{c\},\varnothing \}$ , and $F = \{\{f\}\}$ . We have that

$$\begin{align*}d(\{A,B\},\{C,F\}) = \max(e(A,\{C,F\}), e(B,\{C,F\}),e(C,\{A,B\}),e(F,\{A,B\})). \end{align*}$$

For any x, $d(\{x\},\varnothing ) = 1$ . This implies that $d(\{\{x\},\varnothing \},\{\{y\}\}) = 1$ for any x and y as well. This implies, for instance, that

$$\begin{align*}e(A,\{C,F\}) = \min(d(A,C),d(A,F)) = \min(d(A,C),1) = d(A,C). \end{align*}$$

This together with similar facts for the other three terms implies that

$$ \begin{align*} d(\{A,B\},\{C,F\}) &= \max(d(A,C),d(B,F),d(C,A),d(F,B)) \\ &= \max(d(A,C),d(B,F)). \end{align*} $$

Finally, $d(\{\{x\}\},\{\{y\}\}) = d(x,y)$ and $d(\{\{x\},\varnothing \},\{\{y\},\varnothing \}) = d(x,y)$ for any x and y, so we have that , as required.

Lemma 3.9 means that a sequence of ordered pairs can only converge to an ordered pair and that convergence of sequences of ordered pairs behaves in the expected way. In particular, the class of ordered pairs is closed. (Note that at the moment we don’t know that the collection of ordered pairs is a set, but we will show this later in Corollary 3.15.)

With a little more work we can establish the existence of power sets.

Proof. First note that by basic properties of the Hausdorff metric, the class is necessarily closed. By Lemma 3.3, we know that for any $f \in M$ ,

On the other hand, for any $g \in \mathcal {P}(a)$ , we must have that

by monotonicity of $\inf $ . Therefore,

and so the two quantities are actually equal. Hence $\mathcal {P}(a)$ is a definable class and is coextensive with some element $b\in M$ by Proposition 3.7.

We will write $\mathcal {P}(a)$ for the power set of a.

3.3 Definable functions and replacement

Now that we are confident that ordered pairs exist, the next natural thing to consider is Cartesian products. In order to show that the class is definable and therefore a set, what we would like to be able to do is write a real-valued formula like this:

If we did have this formula, $\varphi (x)$ would of course be the point-set distance from x to the class $a \times b$ . The issue is that the functions $x \mapsto \{x\}$ and $(x,y) \mapsto \{x,y\}$ and the constant $\varnothing $ are not formally part of our logic.

Practically, however, it is commonly understood in the context of discrete logic that it is safe to pretend that certain functions—namely the definable functions—are formally part of the language in the following sense: Given a discrete structure M, a function $f:M^n \to M$ is definable if and only if for every formula $\varphi (\bar {x},y,\bar {z})$ , there is a formula $\psi (\bar {x},\bar {z})$ such that for any $\bar {a},\bar {b} \in M$ , $M \models \varphi (\bar {a},f(\bar {a}),\bar {b})$ if and only if $M \models \psi (\bar {a},\bar {b})$ . It is easy to show that this is equivalent to the graph of f being a definable subset of $M^{n+1}$ .

In continuous logic, a similar thing can be done.

When X is itself definable, it is not too hard to show that f is definable if and only if it is uniformly continuous and its graph is definable (in the sense of Definition 3.5 relative to the max metric on tuples).

While more general statements can be made (see [Reference Yaacov, Alexander Berenstein, Henson and Usvyatsov25, Section 9]), we really only need explicitly definable functions in this article.Footnote ¹⁰

A corollary of Lemma 3.12 is that compositions of explicitly definable functions are explicitly definable. (This is also true of definable functions, but we will not need it.)

It is fairly immediate that the operations we established in Section 3.2 are in fact definable.

Now finally we can return to the question of forming a Cartesian product of two sets. The relevant fact is this:

We will write $a \times b$ for the set whose existence is established in Corollary 3.15.

One thing to note is that the proof of Corollary 3.15 actually establishes that the function $(x,y) \mapsto x \times y$ is explicitly definable as witnessed by the formula

It is occasionally useful to be able to project sets of ordered pairs onto their coordinates. Since the projection function is only partially defined, this is the first time we need the added generality of being able to talk about definable partial functions.

The following facts will also be useful.

3.4 Quotients by discrete equivalence relations

A commonly used operation is passing from an equivalence relation to its set of equivalence classes. We are able to do this for discrete equivalence relations.

We need a small observation.

Lemma 3.20. Let $M \models \mathsf {MSE}$ . If $\varphi (x,y)$ is a discrete equivalence relation on $a \in M$ , then for any ,

is the distance predicate of the $\varphi $ -equivalence class of b.

5.1 Collecting $\varepsilon $ -discrete sets and ‘the’ axiom of infinity

But before we can formalize that, we need to deal with another subtlety we have been ignoring up until now. How do we even know that we can find any uniformly discrete sets? Clearly $\varnothing $ is uniformly discrete, and likewise all hereditarily finite sets are, but it is not clear that the class of hereditarily finite sets is even a set.

What we would like to be able to do is make a formula $\varphi (x)$ that returns , but $\operatorname {\mathrm {dis}}(x)$ is not a continuous function and so cannot possibly be a formula. If $(a_n)_{n \in \mathbb {N}}$ is a Cauchy sequence limiting to a with $a_n \neq a$ for all $n \in \mathbb {N}$ , then $(\{a_n,a\})_{n\in \mathbb {N}}$ will be $d(a_n,a)$ -discrete but no better for every n, yet the limit, $\{a\}$ , will be $1$ -discrete. (This is just the fact that the class of doubletons is not closed again.)

This makes it seem unlikely that we will be able to even approximately collect the r-discrete sets into a class. Nevertheless, we are able to do something nearly as good.

Fix $r> 0$ and consider the formula

Note that if and only if for every , either or $d(y,z) \geq r -\varepsilon $ . If moreover $\varepsilon < \frac {1}{3}r$ , this implies that the formula

$$\begin{align*}E_r(x) = \max(\min(\tfrac{1}{r}(2r-3d(x,y)),1),0) \end{align*}$$

defines a discrete equivalence relation on the elements of a. If a is already r-discrete, then this equivalence relation is equality. For any $\varepsilon> 0$ with $\varepsilon < \frac {1}{3}r$ , let

and let $f_{r} : X_{r} \to M$ be the function that takes a to the set of $E_r$ -equivalence classes of a. By Lemma 3.17 and Proposition 3.21, $f_{r}$ is an explicitly definable partial function. Let $\psi _{r}(x,y)$ be a formula defining it (i.e., for any $a \in X_{r}$ and $b \in M$ , $\psi _{r}^M(a,b) = d(f_{r}(a),b)$ ). Note that $\psi _{r}(x,y)$ does not need any parameters. Note moreover that for any $a \in X_{r}$ , $f_{r}(a)$ is $\frac {1}{3}r$ -discrete.

Definition 5.1. For any $r,\varepsilon> 0$ with $\varepsilon < \frac {1}{3}r$ , we let $\mathsf {Inf}_{r,\varepsilon }$ denote the condition

where means .

We let $\mathsf {Inf}$ denote the collection of conditions $\{\mathsf {Inf}_{1,\varepsilon } : 0<\varepsilon < \frac {1}{3}\}$ .

By the discussion in Section 4, any of the axioms $\mathsf {Inf}_{r,\varepsilon }$ is a sufficient form of the axiom of infinity for the purposes of developing standard mathematics. Nevertheless, we propose the scheme $\mathsf {Inf}$ as a canonical choice for ‘the axiom of infinity’ in the context of $\mathsf {MSE}$ . One objection to this proposal might be that it is a scheme, rather than a single axiom, but as discussed in [Reference Hanson10, Section 6.1], the concept of finite axiomatizability is murky in continuous logic.

For most of the models we construct in Section 7, there is a $1$ -discrete set a for which $\mathsf {Inf}(a)$ holds. This is obviously a more comfortable condition than $M\models \mathsf {Inf}$ , but it is unclear whether it is actually axiomatizable. We could achieve it by adding a constant for some such a, but this is unsatisfying. Thus we have the following question.

General pessimism leads us to believe that the answer to this is no, but we do not see an approach to resolving this question. One candidate might be the existence of the von Neumann ordinal $\omega $ , which is necessarily $1$ -discrete if it exists in an $\omega $ -standard model of $\mathsf {MSE}$ , but it is not clear that any model with a $1$ -discrete infinite set has $\omega $ as an element. It is also not clear whether it’s possible to axiomatize the existence of $\omega $ in a satisfactory way.

5.2 Ordinals

Rather than develop the global structure of cardinals in models of $\mathsf {MSE}$ , we will focus on ordinals. We do this for a couple of reasons. Many of the technical details for cardinals and ordinals are similar but not quite similar enough to develop simultaneously in an expeditious way. Furthermore, more can be said about the structure of ordinals than of cardinals without assuming some form of the axiom of choice.

Note that we will typically use the term well-ordered to mean internally well-ordered. We will use the word ‘externally’ if we wish to emphasize that something is externally well-ordered.

Given a more general sort of linear order, namely a pair $(a,b)$ with a uniformly discrete and $b \sqsubseteq a\times a$ the graph of a linear order, we can find a uniformly discrete chain c such that $(a,b)$ and $(c,{\sqsubseteq }{\upharpoonright } c\times c)$ are internally order-isomorphic. We just need to map each element f of a to the set (i.e., the c-initial segment with largest element f). In this way we can see that uniformly discrete chains are sufficient to represent all uniformly discrete linear order types in models of $\mathsf {MSE}$ .

We denote order types of uniformly discrete well-ordered chains with lowercase Greek letters near the beginning of the alphabet, such as $\alpha $ and $\beta $ . We write to mean that for any a with $\operatorname {\mathrm {otp}}(a) = \alpha $ and any b with $\operatorname {\mathrm {otp}}(b) = \beta $ , a is order-isomorphic to some initial segment of b. We write $\alpha < \beta $ to mean that and $\alpha \neq \beta $ . By a completely standard argument, we have that for any ordinals $\alpha ,\beta \in \operatorname {\mathrm {Ord}}^M$ , either $\alpha < \beta $ , $\beta < \alpha $ , or $\alpha = \beta $ .

It’s important to emphasize that the objects in $\operatorname {\mathrm {Ord}}^M$ are not internal sets in M, but rather externally defined equivalence classes of the equivalence relation $\cong $ . To what extent can we get around this and approximate the class of well-ordered uniformly discrete chains with a set? As is typically the case in set theories with a universal set, something fishy needs to happen with regards to the class of ordinals, on pain of the Burali–Forti paradox. In particular, it is immediate that there cannot be a uniformly discrete set containing representatives of all ordinals of M.

Using techniques similar to those in Section 5.1, we are able to collect representatives of all well-order types occurring below a certain scale. Just as there, we can’t easily form sets that consist solely of r-discrete chains, only things that are in some sense ‘approximate chains.’ We can use a similar trick, however, to turn these into order-isomorphic chains.

Note that $\sigma (a,b) = 0$ if and only if $a \sqsubseteq b$ . Note also that $d(a,b) = \max (\sigma (a,b),\sigma (b,a))$ . Furthermore, $\operatorname {\mathrm {chn}}(a) = 0$ if and only if a is a chain. Let $\varphi _r $ , $E_r$ , $X_r$ , and $f_r$ be defined as they were in Section 5.1.

Define the formula

While $\sigma (x,y)$ measures how close x is to being a subset of y, $\sigma ^\star (x,y)$ measures how close every element of x is to being a subset of every element of y. Lemma 5.6 implies that if $\varphi _r(a) < \frac {1}{3}r$ and $\operatorname {\mathrm {chn}}(a) < \frac {1}{3}$ , then for any $E_r$ -equivalence classes b and c of a, either or . Furthermore, if both of these hold, then $b = c$ .

Let $C_r$ be the class $\{x \in X_r : \operatorname {\mathrm {chn}}(x) < \frac {1}{3}r\}$ . By the above observations, we have that the function $g_r : C_r \to M$ defined by

is explicitly definable (without parameters). (Note that we do not need to take closures as $f_r(a)$ is $\frac {1}{3}r$ -discrete.) Furthermore, it is immediate that $g_r(a)$ is a $\frac {1}{3}r$ -discrete chain for any $a \in C_r$ and if $a \in C_r$ is a chain, then and moreover a and $g_r(a)$ are order-isomorphic as chains.

With this machinery, we are finally in a position to examine the global structure of ordinals in models of $\mathsf {MSE}$ .

Definition 5.7. For any ordinal $\alpha $ of $M \models \mathsf {MSE}$ , we write $s(\alpha )$ for the quantity

$$\begin{align*}\sup\{r> 0 : (\exists~\text{well-ordered}~r\text{-discrete chain}~x\in M)\operatorname{\mathrm{otp}}(x) = \alpha\}. \end{align*}$$

It is easy to see that if , then $s(\alpha ) \geq s(\beta )$ , so for any $M \models \mathsf {MSE}$ , s is a non-increasing map from $\operatorname {\mathrm {Ord}}^M$ to $(0,1]$ . Furthermore, it is always the case that $s(0) = 1$ . By using Hartogs numbers, it is easy to show that for any ordinal $\alpha \in \operatorname {\mathrm {Ord}}^M$ , there is an ordinal $\beta \in \operatorname {\mathrm {Ord}}^M$ of strictly larger cardinality, namely the Hartogs number of $\mathcal {P}(a)$ , where $\operatorname {\mathrm {otp}}(a) = \alpha $ . By an abuse of notation, we’ll write this as $\aleph (\mathcal {P}(\alpha ))$ . Note that by Lemma 4.3 and the fact that the Hartogs number of X always embeds into $\mathcal {P}^3(X)$ , we have that $s(\alpha ) = s(\aleph (\mathcal {P}(\alpha )))$ . This means that if $s(\beta ) < s(\alpha )$ , then $\beta $ has much larger cardinality than $\alpha $ .

We’ll write $\omega ^M$ for the first limit ordinal in M, if it exists. The value of $s(\omega ^M)$ is directly related to the axiom of infinity.

Proof. Fix positive $s < t$ and let $\delta = 1-\frac {s}{t}$ (implying that $s=t(1-\delta )$ ). Assume without loss of generality that $t\delta < \frac {1}{3}r$ . Let

Note that every element of a is an element of $C_r$ . Let . Note that since the collection of chains in M is $\{x \in M : \operatorname {\mathrm {chn}}^M(x) = 0\}$ , it is metrically closed. Hence every element of b is a chain. It is also easy to see that every element of b is $r(1-\delta )$ -discrete (regardless of whether d is an ultrametric).

Let . Note that c is a set in M. Also note that for any $\beta \in \operatorname {\mathrm {Ord}}^M$ , if $s(\beta ) \geq r$ , then some element of c has order type $\beta $ . The equivalence relation $\cong $ is discretely definable on c, so we can form the set . By Lemma 5.9, the set f is $t(1-\delta )$ -discrete (regardless of whether d is an ultrametric) or, in other words, s-discrete. We can find a formula $\varphi (x,y)$ with the property that for any , $\varphi (x,y) \in \{0,1\}$ and $\varphi (x,y) = 0$ if and only . This formula defines a linear order on f which by a standard argument is a well-order. Let

Since f contains representatives of all ordinals $\beta $ with $s(\beta ) \geq r$ , we must have that $\alpha $ is larger than any such $\beta $ . Therefore it cannot be the case that $s(\alpha ) \geq r$ . On the other hand, since f is s-discrete, it follows that $s(\alpha ) \geq s$ , as required.

The last statements in the theorem obviously follow from the rest of it.

The behavior of ultrametric models of $\mathsf {MSE}$ in Theorem 5.10 is reminiscent of the behavior of $\omega $ in models of Cantor–Łukasiewicz set theory, as discovered by Hájek [Reference Montagna17, Theorem 4.17]. In particular, they both exhibit a manifestation of the sorites paradox: an inability to formalize an induction principle of the form

for a real-valued predicate $\varphi (x)$ on some class of ordinals. For , this induction principle cannot hold even for $\omega $ , but, as we will see in Theorem 7.15, models of $\mathsf {MSE}$ can have arbitrarily large standard ordinals. Theorem 5.10 is also of course similar to the non-existence of $\beta $ -models of $\mathsf {NFU}$ , although the mechanism by which models of $\mathsf {NFU}$ are ill-founded is different. One might idly wonder what could happen if we were to restrict excision to stratified formulas.

What is unclear at the moment is the status of non-ultrametric models of $\mathsf {MSE}$ . Theorem 5.10 does not preclude the possibility of $\beta $ -models of $\mathsf {MSE}$ (i.e., models in which $\operatorname {\mathrm {Ord}}^M$ is externally well-founded), but it seems unlikely that they exist. Every model of $\mathsf {MSE}$ we know how to produce contains a set that is an ultrametric model of $\mathsf {MSE}$ , whereby Theorem 5.10 applies. This leaves the following question.

Given the behavior of the models constructed in Section 7, we conjecture that $\mathsf {MSE}$ has no $\beta $ -models and $\{s(\alpha ) : \alpha \in \operatorname {\mathrm {Ord}}^M\}$ is always dense in $(0,1]$ .

Finally, although this is more or less a cosmetic nicety, we would like to show that we can build canonical ‘tokens’ representing well-order types, i.e., elements of M that somehow canonically represent a well-order type $\alpha $ . In $\mathsf {NFU}$ , this is accomplished by taking $\{x : \operatorname {\mathrm {otp}}(x) = \alpha \}$ . In $\mathsf {ZF}$ and $\mathsf {GPK}^+_\infty $ , this is accomplished by taking the von Neumann ordinal of that order type. Neither of these approaches will work for $\mathsf {MSE}$ , so we will have to do something new.

Definition 5.13. Given a uniformly discrete set a, a set $b \sqsubseteq a \times V^M$ , and an element , we write $b[c]$ for the class . For any chain $a \in M \models \mathsf {MSE}$ and any set $b \sqsubseteq a \times V^M$ , the closed chain union of b is

In other words, $\chi (b)$ is the (not necessarily uniformly discrete) chain constructed out of the image of b as a many-valued function on a.

Given a uniformly discrete chain $a \in M \models \mathsf {MSE}$ , the order token of a is the class

So $\operatorname {\mathrm {otok}}(a)$ is roughly speaking the class (which we will shortly show is a set) of all chains that can be realized as a surjective order-preserving image of a. Thinking of $\operatorname {\mathrm {otok}}(a)$ as a canonical token representing the ordinal of a is analogous to thinking of $\{\{a,b\} : a,b \in V\}$ (i.e., the set of sets of size $1$ or $2$ ) as a canonical token representing the cardinal $2$ , as discussed at the beginning of Section 5.

Now of course, given $\operatorname {\mathrm {otok}}(a)$ , we can build a canonical well-ordered chain with the same order type as a, namely

One can show that $x \mapsto \operatorname {\mathrm {ord}}(x)$ is explicitly definable on the class of r-discrete well-ordered chains for any $r> 0$ .

Naturally, as discussed before, we could attempt to do something similar to Definition 5.13 with cardinalities, but without some form of the axiom of choice, we only seem to be able to build tokens representing equivalence classes of the $\approx ^\ast $ relation (where if there is a surjection from some subset of y onto x and $x \approx ^\ast y$ if and ). Specifically, if we define , we then have that $a \approx ^\ast b$ if and only if $\operatorname {\mathrm {ctok}}^\ast (a) = \operatorname {\mathrm {ctok}}^\ast (b)$ for any uniformly discrete a and b. This raises an obvious question.