2. Part-whole and regularity

In this section, we first introduce the general probabilistic setting in which we will work. Throughout this article, we will not assume that the range of a probability measure $\mu$ is contained in the real unit interval $[0,1]$ . The motivation for this is twofold. First, Hájek’s (Reference Hájek2011) “arms race” description of the dialectic between proponents and opponents of regularity involves considering domains and codomains of ever-increasing cardinality. Second, as we shall see later on, the relationship between regularity, totality, and cardinality in the classical setting is muddled by the fact that the standard unit interval $[0,1]$ is Archimedean. We may therefore hope to gain a clearer understanding of the possible obstacles to the existence of a total and regular probability measure in a more general context in which Archimedeanity does not play a role anymore. However, given the importance of probability measures with a range contained in the real interval $[0,1]$ in the literature, we will often point out how the results we discuss apply in this specific case. We start with the following definitions adapted from Pruss (Reference Pruss2013).

Given a generalized probability space, $(\Omega, \rm{\mathfrak{A}},\mu )$ , the generalized $V$ -probability measure $\mu$ is total if $\rm{\mathfrak{A}}=\rm{\mathscr{P}}(\Omega )$ , and it is regular if $\mu (A)=0$ implies $A=\emptyset $ for any $A\in \rm{\mathfrak{A}}$ .

For any ordered field $F$ , the nonnegative elements of $F$ form a generalized probability range. For this reason, we will sometimes consider fields as generalized probability ranges, despite the slight abuse of language. Note that in the classical setting, a finitely additive probability measure is a generalized $\rm{\mathbb{R}}$ -probability measure $(\Omega, \rm{\mathfrak{A}},\mu )$ satisfying the additional condition that $\mu (\Omega )=1$ .

Pruss (Reference Pruss2013) shows a version of the following:

Our first main result (Theorem 3.1) is an improvement of Pruss’s result. In this section, we lay the groundwork for this theorem by establishing a connection between generalized regular probability measures and part-whole–preserving functions. We start by recalling a result that is well known in the literature (e.g., it is stated en passant in Benci et al. [Reference Benci, Horsten and Wenmackers2018], 516) but is worth proving expressly. We first need the following definition:

Proof. First, it is a simple exercise to verify that any generalized $V$ -probability space $(\Omega, \rm{\mathfrak{A}},\mu )$ has the following two properties: $\mu (\emptyset )=0$ , and $\mu (A)\leq \mu (B)$ whenever $A,B\in \rm{\mathfrak{A}}$ and $A\subseteq B$ .

Fix $V, \Omega, \rm{\mathfrak{A}}$ , and $\mu \colon \rm{\mathfrak{A}}\rightarrow V$ . If $A\subsetneq B$ for some $A,B\in \rm{\mathfrak{A}}$ , then finite additivity implies that $\mu (B)=\mu (A)+\mu (B\backslash A)$ . If $\mu$ is regular, then we have $0\lt \mu (B\backslash A)$ , from which it follows that

$\mu \left(A\right)=\mu \left(A\right)+0\lt \mu \left(A\right)+\mu \left(B\backslash A\right)=\mu \left(B\right).$

Hence, $\mu$ preserves part-whole.

Now suppose $\mu$ preserves part-whole, and let $A$ be any nonempty set in $\rm{\mathfrak{A}}$ . Because $\emptyset \subsetneq A$ , by part-whole we have $\mu (\emptyset )\lt \mu (A)$ . Thus, $\mu (A) \neq \mu (\emptyset )=0$ for every nonempty subset $A$ , which means that $\mu$ is regular.□

We now introduce a theorem that will be of great use for our proofs. Its significance for the study of abstraction operators that are used in the context of neologicism has been highlighted by Mancosu and Siskind (Reference Mancosu and Siskind2019) and for issues related to the axiom of choice in second-order logic by Siskind et al. (Reference Siskind, Mancosu and Shapiro2023). The theorem, in its set-theoretic version, is implicit in Zermelo (Reference Zermelo1904) and was brought out explicitly by Kanamori (Reference Kanamori1997).

The theorem can be proved without the axiom of choice, and it can be formalized in a second-order theory with only the additional symbol for $f$ (see Mancosu and Siskind, Reference Mancosu and Siskind2019). Here is a set-theoretic proof that uses transfinite induction:

Proof. Fix a set $X$ and a function $f\,\colon\, \rm{\mathscr{P}}(X)\rightarrow X$ . In order to violate part-whole, we must find two sets $A,B\in \rm{\mathscr{P}}(X)$ such that $A$ is a proper subset of $B$ and $f(A)=f(B)$ . Using transfinite recursion, define a function $G\,\colon\, Ord\rightarrow \rm{\mathscr{P}}(X)$ , where $Ord$ is the class of ordinals, as follows:

$G\left(0\right)=f\left(\emptyset \right);$

$G\left(\alpha \right)=f\left(ran\left(G| \alpha \right)\right),{\rm where\ }{ran}(G| \alpha )=\{G\left(\beta \right)| \beta \lt \alpha \}.$

One easily verifies that for any ordinal $\alpha, G(\alpha )\subseteq X$ . Because $\rm{\mathscr{P}}(X)$ is a set and $Ord$ is a proper class, there must be $\alpha \lt \beta \in Ord$ such that $G(\alpha )=G(\beta )$ . Letting $A=G| \alpha$ and $B=G| \beta$ , this means that $f(A)=G(\alpha )=G(\beta )=f(B)$ . At the same time, we clearly have that $A\subsetneq B$ , which shows that $A$ and $B$ are the required counterexamples to part-whole.□

The connection between the previous applications and the probabilistic setting is the following. Just as in neologicism we are concerned with functions defined by abstraction principles mapping concepts (conceived extensionally as subsets of the domain of individuals) on a certain domain into the domain itself, in many applications of probability theory, we are also mapping a collection of subsets (or the entire power set) of the sample space $\Omega $ into $\Omega $ itself.

The following result is the first immediate application of Zermelo’s (Reference Zermelo1904) result in the context of probability theory:

Theorem 2.7 only uses a special case of Zermelo’s theorem—namely, when the codomain of the function $f$ is a subset of the unit interval. As we shall now see, the generality of Zermelo’s theorem allows for a much more general impossibility result regarding the existence of regular probability measures.

3. Zermelo’s theorem and generalized probability measures

Our first generalization of Theorem 2.7 allows us to consider a much wider class of sample spaces and probability ranges.

Proof. Suppose toward a contradiction that $\mu$ is a regular generalized $V$ -probability measure for $\Omega $ . Because $| \Omega | \geq | V|$ , there is a surjection $f\,\colon\, \Omega \rightarrow V$ . Consider the inverse image lift $f_{\rm{*}}\,\colon\, \rm{\mathscr{P}}(V)\rightarrow \rm{\mathscr{P}}(\Omega )$ of the surjection $f$ , given by $f_{\rm{*}}(A)=\{y\in \Omega | f(y)\in A\}$ for any $A\subseteq V$ . One easily verifies the following:

• $f_{\rm{*}}(\emptyset )=\emptyset, f_{\rm{*}}(A)\subseteq f_{\rm{*}}(B)$ whenever $A,B\subseteq V$ and $A\subseteq B$ ; and

• for any $A,B\subseteq V, f_{\rm{*}}(A\cup B)=f_{\rm{*}}(A)\cup f_{\rm{*}}(B), f_{\rm{*}}(A\cap B)=f_{\rm{*}}(A)\cap f_{\rm{*}}(B)$ ;

• $f_{\rm{*}}$ is an injection.

Now consider the composition $\nu =\mu \circ f_{\rm{*}}\,\colon\, \rm{\mathscr{P}}(V)\rightarrow V$ . We then verify that $\nu$ is a regular generalized $V$ -probability measure. Indeed:

• For any $A\subseteq V, \nu (A)=\mu (f_{\rm{*}}(A))\geq 0$ by nonnegativity of $\mu$ .

• For any $A,B\subseteq V$ such that $A\cap B=\emptyset $ :

$\nu \left(A\cup B\right)=\mu \left(f_{\rm{*}}\left(A\cup B\right)\right)=\mu \left(f_{\rm{*}}\left(A\right)\cup f_{\rm{*}}\left(B\right)\right)=\mu \left(f_{\rm{*}}\left(A\right)\right)+\mu \left(f_{\rm{*}}\left(B\right)\right)$

by finite additivity of $\mu$ , and because $f_{\rm{*}}(A)\cap f_{\rm{*}}(B)=f_{\rm{*}}(A\cap B)=\emptyset $ . Hence,

$\nu \left(A\cup B\right)=\nu \left(A\right)+\nu \left(B\right),$

which shows that $\nu$ is finitely additive.

• For any $A\subseteq V, (A)=0$ implies that $\mu (f_{\rm{*}}(A))=0$ . Because $\mu$ is regular, this means that $f_{\rm{*}}(A)=\emptyset $ . Because $f_{\rm{*}}$ is injective and $f_{\rm{*}}(A)=f_{\rm{*}}(\emptyset )=\emptyset $ , it follows that $A=\emptyset $ , which establishes that $\nu$ is regular.

By Lemma 2.5, this means that there is a part-whole–preserving function $\mu \,\colon\, \rm{\mathscr{P}}(V)\rightarrow V$ , which contradicts Zermelo’s theorem. Hence, there is no generalized $V$ -probability measure $\mu \,\colon\, \rm{\mathscr{P}}(\Omega )\rightarrow V$ .□

Theorem 3.1 is a clear generalization of Pruss’s (Reference Pruss2013) result. By appealing to Zermelo’s theorem, we weakened the assumption that $\Omega $ has size strictly greater than $V$ to merely assuming the existence of a surjection from $\Omega $ to $V$ .

Moreover, Pruss’s (Reference Pruss2013) result only shows that there must be at least one failure of regularity for any $V$ -probability measure. Using Theorem 3.1, however, we can actually show that there must be many failures of regularity, even in the generalized setting.

Finally, we conclude this section by discussing in some detail the special case of real-valued finitely additive probability measures. As an immediate corollary of Theorem 3.1, we obtain the following:

Corollary 3.3 is a weaker form of a well-known result establishing that there can be no finitely additive regular probability measure from a set $\Omega $ into $[0,1]$ when $\Omega $ is uncountable. The argument has been presented neatly by both Hájek (Reference Hájek2003) and Williamson (Reference Williamson2007). Let $\Omega $ be a set, and suppose that $\mu$ is a finitely additive probability measure into $[0,1]$ and defined on all singletons. For any natural number $n$ , let $A_{n}$ be the set of all $x\in \Omega $ such that $\mu \left( {\left\{ x \right\}} \right) \ge {1 \over n}$ . By finite additivity, each $A_{n}$ must be finite, which means that there can only be countably many singletons mapped to a nonzero value in $[0,1]$ . If $\Omega $ is uncountable, then this means that uncountably many singletons must be mapped to $0$ .

The Hájek–Williamson argument has three advantages. The first one is that it shows that the failure of regularity must occur in many places because uncountably many singletons must be mapped to $0$ . By contrast, Corollary 3.3 only guarantees that regularity must fail in some place, which is enough to conclude that any standard probability measure must assign probability $0$ to $\{x\}$ for some $x$ , but nothing more. Here, however, it is easy to see that one can adapt the proof of Lemma 3.2 in a straightforward way so as to obtain continuum many failures of regularity in the case of a probability measure defined on the full powerset of a sample space of size continuum. Second, although our application of Zermelo’s theorem requires the sample space $\Omega $ to have size continuum, one only needs $\Omega $ to be uncountable for the Hájek–Williamson argument to go through. In the absence of the continuum hypothesis, the Hájek–Williamson result is therefore stronger than ours. Finally, the Hájek–Williamson argument only requires the probability measure to be defined on all singletons,Footnote ⁷ whereas our results assume that the probability measure is total. However, the last two differences between the two results are connected to a third significant one: the Hájek–Williamson argument relies in an essential way on the fact that the real interval $[0,1]$ is Archimedean. By finite additivity, only finitely many singletons can have probability greater than ${1 \over n}$ for a fixed natural number $n$ , which means that uncountably many singletons must have a value smaller than ${1 \over n}$ for any $n$ . But this implies that uncountably many singletons must have probability $0$ only because $0$ is the unique element smaller than ${1 \over n}$ for any $n$ (i.e., $[0,1]$ is Archimedean). The Hájek–Williamson argument does not say anything about the failure of regularity in a non-Archimedean field, which contrasts with the generality of Theorem 3.1. We will return to the issue of Archimedeanity in the next section, in which we investigate to what extent totality is necessary to establish the impossibility results mentioned so far.

4. Relaxing totality

In this section, we investigate the consequences of relaxing the requirement that the algebra of events must be the entire powerset of $\Omega $ for the existence of regular measures. Let us start with an easy result.

In the next section (Theorem 5.5), we will recall existing results in the literature establishing that for any set $\Omega $ there is a hyperreal field $V$ and a regular generalized $V$ -probability measure defined on $\rm{\mathscr{P}}(\Omega )$ . This entails that the assumption in the statement of Lemma 4.1 can in fact always be satisfied.

For now, let us investigate some consequences of this lemma for countable sample spaces and real-valued probability measures. In order to do this, we start by recalling a classical result of measure theory (e.g., it is a special case of theorem 2.5 in Horn and Tarski [Reference Horn and Tarski1948]):

Proof. Let us first show this in the case $\Omega =\rm{\mathbb{N}}$ . Recall that any real number in the interval $[0,1]$ has a binary expansion; that is, it can be written as a (possibly infinite) sequence of negative powers of 2. For example, the rational number $0.625$ can be written as $0.5+0.125=1\cdot 2^{-1}+0\cdot 2^{-2}+1\cdot 2^{-3}$ . Conversely, any countable sequence $s$ of $0s$ and $1s$ determines the binary expansion of some real number $r(s)\in [0,1]$ . For any subset $U$ of $\rm{\mathbb{N}}$ , let $\mu _{0}(U)=r(\chi _{U})$ ; that is, let $\mu _{0}(U)$ be the real number whose binary expansion is determined by the characteristic function of $U$ . Formally, this means that for any $U\subseteq \rm{\mathbb{N}}$ :

$\mu _{0}\left(U\right)=\displaystyle\sum _{i\in \rm{\mathbb{N}}}\chi _{U}\left(i\right)\cdot 2^{-i}.$

For example, $\mu _{0}(\{1,3\})=1\cdot 2^{-1}+0\cdot 2^{-2}+1\cdot 2^{-3}=0.625$ . Then observe the following:

• $\mu _{0}(U)=0$ iff $\chi _{U}$ is constantly $0$ iff $U=\emptyset $ .

• Whenever $U\cap V=\emptyset $ , we have that:

$\mu _{0}\left(U\right)+\mu _{0}\left(V\right)=\sum _{i\in \rm{\mathbb{N}}}\chi _{U}\left(i\right)\cdot 2^{-i}+\sum _{i\in \rm{\mathbb{N}}}\chi _{V}\left(i\right)\cdot 2^{-i}$

$\hskip 2.2pc=\sum _{i\in N}\left(\chi _{U}\left(i\right)+\chi _{V}\left(i\right)\right)\cdot 2^{-i}$

$\hskip 3.1pc=\sum _{i\in N}\max \left(\chi _{U}\left(i\right),\chi _{V}\left(i\right)\right)\cdot 2^{-i}$

$\hskip -1.2pc=\sum _{i\in N}\chi _{U\cup V}\cdot 2^{-i}$

$\hskip -2.5pc=\mu _{0}\left(U\cup V\right),$

which shows that $\mu _{0}$ is finitely additive (in fact, the same argument shows that it is $\sigma$ -additive).

Hence, $\mu _{0}$ is a regular finitely additive probability measure defined on the whole of $\rm{\mathscr{P}}(\rm{\mathbb{N}})$ . Now if $\Omega $ is any countable set, there is a surjection $f\,\colon\, \rm{\mathbb{N}}\rightarrow \Omega $ , which can be lifted to an injection $f_{\rm{*}}\,\colon\, \rm{\mathscr{P}}(\Omega )\rightarrow \rm{\mathscr{P}}(\rm{\mathbb{N}})$ , as in the proof of Theorem 3.1. But then $\mu _{0}\circ f_{\rm{*}}\,\colon\, \rm{\mathscr{P}}(\Omega )\rightarrow [0,1]$ is also a regular finitely additive probability measure defined on $\rm{\mathscr{P}}(\Omega )$ .□

Combined with a similar construction as in Lemma 4.1, we get the following as an immediate corollary of Lemma 4.2:

Moreover, Lemmas 4.1 and 4.2 also have some immediate consequences regarding the interaction of probability theory and computability theory. For example, it is well known that there are only countably many recursive subsets of $\omega$ and that they form a Boolean subalgebra $\rm{\mathbb{A}}$ of $\rm{\mathscr{P}}(\omega )$ . By Lemma 4.1, we may consider the subfield $V\rm{'}$ of $\rm{\mathbb{R}}$ generated by the range of the restriction of $\mu _{0}$ to $\rm{\mathbb{A}}$ , where, again, $\mu _{0}$ is the function defined in the proof of Lemma 4.2. In fact, it is easy to verify that $V\rm{'}$ is exactly the field of all recursive reals in the interval $[0,1]$ . This has the following consequence:

In other words, when restricting our attention to a countable subalgebra of the powerset of the recursive reals in the interval $[0,1]$ , we are able to escape the conclusion of Zermelo’s theorem. Of course, this is only because $\rm{\mathfrak{A}}$ is not the full powerset of the set of recursive reals because there are uncountably many subsets of the recursive reals. Let us also note that the regular probability measure thus constructed is not itself recursive, nor is $\rm{\mathfrak{A}}$ the algebra of recursive subsets of $\rm{\mathbb{R}}\rm{'}$ .

The previous results establish that relaxing the assumption of totality may sometimes allow for the existence of regular probability measures. One is therefore led to wonder whether totality is in fact a necessary condition for the conclusion of Zermelo’s theorem to apply. The proof of Zermelo’s original theorem certainly seems to make essential use of the fact that the function $f\,\colon\, \rm{\mathscr{P}}(X)\rightarrow X$ that ends up violating part-whole is defined on all arbitrary subsets of $X$ . However, the obstacle to the existence of regular probability measures is not totality per se because it is sufficient that the algebra of events be merely isomorphic to the full powerset of the sample space, as the following straightforward lemma shows:

As a direct consequence of Lemma 4.5, we can now prove the following impossibility theorem for algebras of events that are far from being the full powerset of the sample space:

This result establishes that totality is not, after all, a necessary condition for the conclusion of Zermelo’s theorem to hold. Indeed, to guarantee that there can be no regular generalized $V$ -probability measure defined on a subalgebra $\rm{\mathfrak{A}}$ of $\rm{\mathscr{P}}(\Omega )$ for some set $\Omega $ of size $| V|$ , it is enough to assume that $\rm{\mathfrak{A}}$ is isomorphic to $\rm{\mathscr{P}}(\Omega )$ , which, as corollary 4.6 shows, does not coincide with $\rm{\mathfrak{A}}$ being the full powerset of $\Omega $ . Moreover, the real obstacle to the existence of a regular function also does not lie in the constraints put on the cardinality of the algebra of events, as the following result shows:

Proof. Let $V$ be a countable non-Archimedean field.Footnote ⁸ Fix a positive infinitesimal $\varepsilon \in V$ . By definition, any element in $Fin(\Omega )$ is either a finite subset of $\Omega $ or the complement of such a finite subset. Define $\mu \,\colon\, Fin(\Omega )\rightarrow V$ by letting $\mu (A)=\varepsilon | A|$ if $A$ is finite, and $(A)=1-\varepsilon | \Omega \backslash A|$ otherwise. Let us show that $\mu$ is the required measure. Suppose $A,B\in Fin(\Omega )$ are such that $A\cap B=\emptyset $ . We distinguish two cases:

• Case 1: Both $A$ and $B$ are finite. In this case we have that $A\cup B$ is finite and $| A\cup B| =| A| +| B|$ , so

$\mu \left(A\cup B\right)=\varepsilon \left| A\cup B\right| =\varepsilon \left(\left| A\right| +\left| B\right| \right)=\varepsilon \left| A\right| +\varepsilon \left| B\right| =\mu \left(A\right)+\mu \left(B\right).$

• Case 2: Either $A$ or $B$ is infinite. Without loss of generality, assume $A$ is infinite. Note that this means that $\Omega \backslash A$ is finite. Because $A\cap B=\emptyset $ , we have that $B\subseteq \Omega \backslash A$ , so $B$ is also finite. By simple finite cardinality reasoning, we have that $| \Omega \backslash A| =| B| +| (\Omega \backslash A)\cap (\Omega \backslash B)|$ . Moreover, we have that $\Omega \backslash (A\cup B)=(\Omega \backslash A)\cap (\Omega \backslash B)$ . Because $A$ and $A\cup B$ are infinite but $B$ is finite, it follows that $\mu (A)=1-\varepsilon | \Omega \backslash A|, \mu (A\cup B)=1-\varepsilon | (\Omega \backslash A)\cap (\Omega \backslash B)|$ , and $\mu (B)=\varepsilon | B|$ . Now we compute:

$\left| \Omega \backslash A\right| =\left| B\right| +\left| \left(\Omega \backslash A\right)\cap \left(\Omega \backslash B\right)\right| \Leftrightarrow$

$\left| B\right| =\left| \Omega \backslash A\right| -\left| \left(\Omega \backslash A\right)\cap \left(\Omega \backslash B\right)\right| \Leftrightarrow$

$\varepsilon \left| B\right| =\varepsilon \left| \Omega \backslash A\right| -\varepsilon \left| \left(\Omega \backslash A\right)\cap \left(\Omega \backslash B\right)\right| \Leftrightarrow$

$1-\varepsilon \left| \Omega \backslash A\right| +\varepsilon \left| \Omega \backslash B\right| =1-\varepsilon \left| \left(\Omega \backslash A\right)\cap \left(\Omega \backslash B\right)\right| \Leftrightarrow$

$\mu \left(A\right)+\mu \left(B\right)=\mu \left(A\cup B\right).$

It follows that $\mu$ is a $V$ -probability measure. Moreover, by construction, $\mu$ is clearly regular.□

Lemma 4.7 shows that the cardinality of the algebra of events does not determine by itself how big the codomain of a regular generalized probability measure must be. Indeed, a countable codomain will suffice for a sample space $\Omega $ , provided that the algebra of events under consideration is generated by the singletons. But such an algebra has the same cardinality as $\Omega $ , which shows that a countable codomain is enough to find regular measures for arbitrarily large algebras.

We conclude this section by connecting this last result with our discussion of Archimedeanity in the previous section:

5. Regularity, cardinality, and hyperreal fields

In this final section, we connect our investigations on the relationship between totality, regularity, and cardinality to the nonstandard approach to probability and to hyperreal extensions of $\rm{\mathbb{R}}$ . We will limit ourselves to the model-theoretic approach to nonstandard analysis because our interest in the cardinality of domains and codomains of regular probability measures does not align with an axiomatic approach à la Nelson. We start by recalling the framework of enlargements that is central in Robinsonian nonstandard analysis.Footnote ⁹

Definition 5.2 An enlargement is a map $ ^{*}\,\colon\, \rm{\mathbb{U}}(\rm{\mathbb{X}})\rightarrow \rm{\mathbb{U}}(\rm{\mathfrak{X}})$ between universes such that:

• $\rm{\mathbb{X}}\subsetneq \rm{\mathfrak{X}}$ , and $^{\rm{*}}x=x$ for any $x\in \rm{\mathbb{X}}$ ;

• $\rm{\mathfrak{X}}=^{\rm{*}}\rm{\mathbb{X}}$ and $^{\rm{*}}\emptyset =\emptyset $ ;

• (Transfer Principle) for any first-order formula $\varphi (x_{1},..,x_{n})$ in the language of set theory and any $a_{1},\ldots,a_{n}\in \rm{\mathbb{U}}(\rm{\mathbb{X}})$ :

$\rm{\mathbb{U}}\left(\rm{\mathbb{X}}\right)| =\varphi \left(a_{1},\ldots,a_{n}\right)\Leftrightarrow ^{\rm{*}}\rm{\mathbb{U}}\left(\rm{\mathbb{X}}\right)| =\varphi (^{\rm{*}}a_{1},\ldots,^{\rm{*}}a_{n});\rm{and}$

• (Saturation Principle) whenever $A\in \rm{\mathbb{U}}(\rm{\mathbb{X}})$ is collection of sets such that for any $a_{1},\ldots,a_{n}\in A, \cap _{i\leq n}a_{i}\neq \emptyset $ , there is $b\in \rm{\mathbb{U}}(\rm{\mathbb{X}})$ such that $b\in \cap _{i\leq n}^{\rm{*}}a_{i}$ .

Whenever $^{\rm{*}}\,\colon\, \rm{\mathbb{U}}(\rm{\mathbb{R}})\rightarrow \rm{\mathbb{U}}(\rm{\Re })$ is an enlargement, $\rm{\Re }$ is a hyperreal extension of $\rm{\mathbb{R}}$ , meaning that it satisfies the same first-order theory in the language of fields as $\rm{\mathbb{R}}$ , but it contains infinitely small and infinitely large numbers. In particular, one obtains a nonstandard unit interval $[0,1]_{\rm{\Re }}=^{\rm{*}}[0,1]_{\rm{\mathbb{R}}}$ . Using a nonstandard framework similar to enlargements, Bernstein and Wattenberg (Reference Bernstein and Wattenberg1969) showed the following:

One must exert some care in distinguishing the hyperreal field $\rm{\Re }$ used by Bernstein and Wattenberg (Reference Bernstein and Wattenberg1969) from the simplest type of hyperreal extensions obtained by taking an ultrapower of the reals modulo a nonprincipal ultrafilter on $\omega$ . Indeed, if $R$ is such an ultrapower, then

$| [0,1]_{R}\left| =\right| \{f| f\,\colon\, \omega \rightarrow [0,1]_{\rm{\mathbb{R}}}\}\left| =\right| [0,1]_{\rm{\mathbb{R}}}| ^{{\aleph _{0}}}=(2^{{\aleph _{0}}})^{{\aleph _{0}}}=2^{{\aleph _{0}}},$

and hence $[0,1]_{R}$ has the same size as $\rm{\mathbb{R}}$ . But then it follows directly from Theorem 3.1 that there can be no total and regular probability measure from $\rm{\mathscr{P}}(\rm{\mathbb{R}})$ into $[0,1]_{R}$ . Thus, the hyperreal field $\rm{\Re }$ considered by Bernstein and Wattenberg cannot be isomorphic to such a hyperreal extension $R$ because it must have size strictly greater than $2^{{\aleph _{0}}}$ .

Another important aspect of this nonstandard setting is the distinction between internal and external objects. Given an enlargement $^{\rm{*}}\,\colon\, \rm{\mathbb{U}}(\rm{\mathbb{X}})\rightarrow \rm{\mathbb{U}}(\rm{\mathfrak{X}})$ , an element $a\in \rm{\mathbb{U}}(\rm{\mathfrak{X}})$ is internal if $a\in \cup *\rm{\mathbb{U}}(\rm{\mathbb{X}})=\cup \{^{*}\it b| b\in \rm{\mathbb{U}}(\rm{\mathbb{X}})\}$ , and external otherwise. The distinction is crucial for many applications of nonstandard analysis because internal objects generally behave in a much more tractable fashion than external objects. An important example of internal objects is the internal subsets $\rm{\mathscr{P}}_{\rm{Int}}(^{\rm{*}}\it A)$ of the image $^{\rm{*}}A$ of some standard object $A$ . In particular, one has that for any $A\in \rm{\mathbb{U}}(\rm{\mathbb{X}})$ ,

$^{\rm{*}}\rm{\mathscr{P}}\left(\it A\right)=\rm{\mathscr{P}}_{\rm{Int}}(^{\rm{*}}\it A).$

In the context of an enlargement $^{\rm{*}}\,\colon\, \rm{\mathbb{U}}(\rm{\mathbb{R}})\rightarrow \rm{\mathbb{U}}(\rm{\Re })$ , one may therefore be interested in considering both internal and external entities when discussing the existence of total and regular probability measures on the nonstandard unit interval $[0,1]_{\rm{\Re }}$ . The following is straightforward to verify:

It is worth mentioning that Lemma 5.4 does not address the issue of whether there can be an external $[0,1]_{\rm{\Re }}$ -valued regular probability measure that is defined on all the internal subsets of $[0,1]_{\rm{\Re }}$ . Note in particular that the results of Section 4 do not apply here because $\rm{\mathscr{P}}_{\rm{Int}}([0,1]_{\rm{\Re }})$ has the same size as the full powerset of $[0,1]_{\rm{\Re }}$ , but it is not a complete algebra, and thus it is not isomorphic to $\rm{\mathscr{P}}([0,1]_{\rm{\Re }})$ .Footnote ¹⁰ We leave this technical issue as an interesting open problem for the time being and limit ourselves to pointing out that nonstandard frameworks obtained as limits of enlargements might be a promising approach toward a positive answer to this problem.Footnote ¹¹

We conclude this section by discussing the connection between our results and an alternative approach to probability theory inspired by nonstandard analysis—namely, the non-Archimedean probability (NAP) theory developed by Benci et al. (Reference Benci, Horsten and Wenmackers2013, Reference Benci, Horsten and Wenmackers2018). The main advantage of NAP over classical probability theory is that one can construct, for any set $\Omega $ , a total and regular probability measure on $\rm{\mathscr{P}}(\Omega )$ whose codomain is a reasonably small hyperreal extension of $\rm{\mathbb{R}}$ . We briefly review their construction here.

Let $\Omega $ be an infinite set. We write $\rm{\mathscr{P}}_{\rm{Fin}}(\Omega )$ for the set of all finite subsets of $\Omega $ , and we write $^{\Omega }\rm{\mathbb{R}}$ for the ring of functions from $\rm{\mathscr{P}}_{\rm{Fin}}(\Omega )$ into $\rm{\mathbb{R}}$ , where the operations $+$ and $\cdot$ and the order $\lt$ are defined pointwise. A fine ideal on $^{\Omega }\rm{\mathbb{R}}$ is a maximal ideal containing the set

$\{\varphi \,\colon\, \rm{\mathscr{P}}_{\rm{Fin}}\left(\Omega \right)\rightarrow \rm{\mathbb{R}}| \exists A\subseteq \rm{\mathbb{R}~ }\forall B\supseteq A\,\colon\, \varphi \left(B\right)=0\}.$

Given a fine ideal $I$ , the quotient $^{\Omega }\rm{\mathbb{R}}/I$ is a hyperreal field that extends $\rm{\mathbb{R}}$ and has size

$| ^{\Omega }\rm{\mathbb{R}}\left| =\right| \rm{\mathbb{R}}| ^{\left| \rm{\mathscr{P}}_{\rm{Fin}}\left(\Omega \right)\right| }=(2^{{\aleph _{0}}})^{\left| \Omega \right| }=2^{\left| \Omega \right| }.$

Benci et al. (Reference Benci, Horsten and Wenmackers2013) prove the following.

Benci et al. (Reference Benci, Horsten and Wenmackers2013) call such functions NAP measures and show that they exhibit a number of attractive properties. In particular, they satisfy a notion of additivity that is arguably a generalization of countable additivity in the non-Archimedean context. Combined with Theorem 3.1, this yields the following result regarding the interplay between cardinality, regularity, and totality for hyperreal fields:

Finally, let us note that under the assumption of GCH, Theorem 5.6 yields a complete answer to the question of the compatibility between regularity and totality as raised in Section 1.

We conclude by briefly discussing the significance of our results to the debate regarding regularity in probability theory. In short, we see our results as confirming that the debate has been essentially well posed by Hofweber (Reference Hofweber2014b) and Hájek (Reference Hájek2011). On the one hand, Theorem 5.7 can be viewed as the mathematical fact underlying Hájek’s (Reference Hájek2011) “arms race.” Indeed, as a consequence of Zermelo’s theorem, for any possible range of probability values $V$ , any regular probability measure defined on the full powerset of $V$ must take its values in a range of cardinality strictly greater than $V$ . In that sense, no hyperreal field can be “large enough” to play the role of the reals in an alternative to Kolmogorov probability theory that would satisfy the regularity constraint. At the same time, Hofweber is correct in saying that the cardinality of the codomain is the only substantial obstacle to the existence of a regular probability measure for any sample space. In fact, under GCH, it is necessary and sufficient to take the successor cardinal of the size of the domain as the size of the codomain of such a regular probability measure. To make this point differently, this means that, at least under GCH, the existence theorems noted by Hofweber (Reference Hofweber2014a) and Benci et al. (Reference Benci, Horsten and Wenmackers2018) are the best possible. Interestingly, this also has a consequence for a slightly different question, regarding whether any standard probability function can be approximated by a regular function defined on a hyperreal field.Footnote ¹² Hofweber and Schindler (Reference Hofweber and Schindler2016) give a positive answer to this question. Given a standard probability space $(\Omega, \rm{\mathfrak{A}},\overline{\mu })$ , they show that $\overline{\mu }$ can be approximated by a regular probability measure $\mu$ (in the sense that $\mu (U)$ is infinitesimally close to $\overline{\mu }(U)$ for any $U\in \rm{\mathfrak{A}}$ ) defined on a hyperreal field $V$ of size at most $2^{| \Omega | }$ . As an easy consequence of Theorem 5.7, we have that, under GCH, the cardinality bound obtained by Hofweber and Schindler is the best possible. In other words, this means that from the point of view of the cardinality of the range of probabilities, the existence problem (i.e., whether there exists a regular probability measure) and the approximation problem (i.e., whether any standard probability function can be approximated by a regular one) are equivalent.

Finally, let us make one final remark on the issue. As the results in Section 4 show, a lot hinges on the particular structure of the algebra of events under consideration. If one agrees with Hájek (Reference Hájek2011) that the regularity constraint implicitly entails a totality constraint, then the “arms race” is indeed the only option. But if one only wants to consider events that are finite or cofinite sets of outcomes, then, as Lemma 4.7 shows, any countable non-Archimedean field would suffice. We take this as evidence that there is still much to explore regarding the interplay between regularity, totality, and cardinality in probability theory.