Beyond a Naive Absolute Infinite

1. Introduction

In a letter to Dedekind, Cantor called his absolute infinite $\Omega$1 an inconsistent, absolutely infinite multiplicity and he associated it with God (Cantor [7], Thomas-Bolduc [29]). The absolute infinite is sometimes used in theology (Steinhart [26], Blondé [3]), but can be used for various other purposes too: in metaphysics and modal realism to describe the size of Lewis’ ([18]) logical space or the plenitude (Blondé [2]), in computer science to have the ultimate oracle (Burgin [6]) or ordinal2 machine (Koepke [16]), in epistemology to formulate omniscience (Heylen [14]), and in the philosophy of mathematics to have a maximal entity (Sutto [27]).

Cantor’s idea continues to provoke controversy. Welch and Horsten ([31]) review Cantor’s conception of the set-theoretic universe as a completed infinity and prefer it above Zermelo’s ([33]) conception, because Cantor’s universe includes the modern large cardinals via reflection principles. Livadas ([21]) also discusses Cantor’s absolute infinite in light of the modern large cardinals and argues that it is proof-theoretically unattainable. Gutschmidt and Carl ([10]) maintain that the foundational problem with Cantor’s absolute infinite calls for humility, rather than negative theology.

Set-theoretic potentialism, which has garnered significant attention in the last decade, is the view that the process of set formation is incompletable or inexhaustible and that the classical universe of sets V and the absolute infinite $\Omega$ cannot be fully captured or defined as actual, completed totalities (Zermelo [34], Putnam [24], Parsons [23], Hamkins [12], Linnebo [19], Hellman and Shapiro [13], Linnebo and Shapiro [20], Brauer et al. [4], Sutto [27]). Height potentialism can be distinguished from width potentialism. Height potentialism is the claim that certain large classes, such as the class of all natural numbers or the class of all sets, are not actual, but only potential classes. Width potentialism, on the other hand, asserts that there are no privileged axiomatic truths and that there is, therefore, no privileged universe of sets. Instead, there is a wide set-theoretic multiverse of potential multiverse-universes that are equally legitimate (Hamkins [12], Scambler [25], Meadows [22], Gorbow [9]). For example, a multiverse-universe in which the Continuum Hypothesis (CH) is true is not better or worse than one in which CH is false.

In spite of this, the aim of this paper is to show that an absolute infinite $\Omega$ can be proven to exist in a theory that can interpret the notion ‘theory’. This counters height potentialism. The next section introduces the axiom, definitions, and theorems that relate to the absolute infinite. In Section 3, the theorems are proven on the basis of the axiom and the definitions. After that, in Section 4, a range of objections to the existence of $\Omega$ is rebutted. At last, the conclusions follow in Section 5.

2. Toward an Absolute Infinite

2.1. Class Ordinals and Class Cardinals

Class ordinals and class cardinals are defined first as straightforward generalizations of ordinals and cardinals, which will be called set ordinals and set cardinals for clarity. The rationale for this generalization is to create terms to refer to the absolute infinite $\Omega$, namely the proper class ordinal and the proper class cardinal, and unambiguous terms to refer to set ordinals and set cardinals. These terms are used directly in the axiomatization of $\Omega$ in Section 2.2. A set $\alpha$ is a set ordinal if and only if (henceforth iff) (Kunen [17], p. 16):

• $\alpha$ is a transitive set.
• ($\alpha$, ∈) is a well-ordering.

The Burali-Forti ([5]) paradox shows that the collection of all set ordinals Ord cannot be a set (Kunen [17], p. 17). If such a collection would be a set, it would exceed itself in size. However, a collection that is too large to be a set, such as Ord, can still be a class, namely a proper class (von Neumann [30]). Every set is a class, namely a class that is an element of another class, but not every class is a set. In particular, proper classes, which are classes that are not an element of any other class, are not sets. For example, the class-level axioms of NBG (von Neumann–Bernays–Gödel) can be used when a theory about sets and classes is needed (von Neumann [30]). With this notion of classes, class ordinals can be defined as an extension of set ordinals. A class C is a class ordinal iff:

• C is a transitive class.
• (C, ∈) is a well-ordering.

A class C is transitive iff whenever

$$x\in C\wedge y\in x\to y\in C$$

(1)

($C,\in$), with C some class, is a well-ordering iff every non-empty set $s\subseteq C$ has an ∈-least element:

$$\forall s\subseteq C,s\ne \varnothing :\exists x\in s,\forall y\in s,x\ne y\to x\in y$$

(2)

This definition of class ordinal is a straightforward NBG-class-level generalization of the von Neumann definition of (set) ordinal. By the usual ordinal analysis (Kunen [17], p. 16), if C is a class ordinal and $x\in C$, then x is a set that is a set ordinal. A proper class ordinal is defined as any class ordinal that is not a set. Like any class, any class ordinal C is either a set or a proper class. If C is a set, then by definition of class ordinal it is transitive and well-ordered by ∈, and therefore C is a set ordinal. If C is not a set, then by definition it is a proper class ordinal. Hence we have a dichotomical ontology for class ordinals: every class ordinal is either a set ordinal or a proper class ordinal.

Just as for set ordinals in Kunen ([17], p. 17), the intersection of two class ordinals is again a class ordinal. In NBG, the intersection of two classes is again a class by class comprehension, so for class ordinals $C_1$ and $C_2$, the intersection $C:=C_1\cap C_2$ is a class. It is transitive because any element of C is in both $C_1$ and $C_2$, which are transitive. At last, ∈ well-orders C because any non-empty subclass of C is also a subclass of $C_1$ and $C_2$, which are well-ordered by ∈.

Another result that is paralleled by set ordinals in Kunen ([17], p. 18) is that a class ordinal cannot be a proper subclass of another class ordinal unless it is an element of it: $C_1⊊C_2\to C_1\in C_2$. This can be proven by showing that $C_1$ is the ∈-least element of the non-empty class $C_2\setminus C_1$. From this it follows that every class ordinal is an initial segment of any greater class ordinal and that we have trichotomical comparability with respect to ∈ between any pair of class ordinals: exactly one of $\alpha \in \beta$, $\alpha =\beta$, or $\beta \in \alpha$, holds.

The proper class Ord is a proper class ordinal because it is transitive and well-ordered by ∈. Given that a proper class ordinal $C_p$ cannot be an element of any class, only the option $C_p=$ Ord remains in the trichotomical comparison. Consequently, Ord is the unique proper class ordinal in its theory. Ord is – again, in its theory – also the supremum of all set ordinals and it is maximal: no class ordinal in its theory exceeds it.

The notions ‘least’ and ‘maximal’ refer to the order ∈ imposes: $\alpha \in \beta$ iff $\alpha <\beta$. The successor $\alpha +1$ of a set ordinal $\alpha$ is defined as $S\left(\alpha \right)=\alpha \cup \left\{\alpha \right\}$ (Kunen [17], p. 18). By the usual ordinal analysis, the successor of a set ordinal is again a set ordinal. Ord cannot have a successor because Ord, being a proper class, cannot be an element of any class, and hence no class of the form Ord ∪{Ord} exists.

Because of the uniqueness and the maximality of proper class ordinals in a given theory, if the absolute infinite $\Omega$ provably exists as a class ordinal in some consistent class theory, it must be equal to a proper class ordinal. Class cardinals can be introduced in a similar way, such that we also have a proper class cardinal that is maximal and unique in its theory.

2.2. The Absolute Infinite: Axiom, Meta-Theory, Definition, and Theorems

This section provides the essence of the paper: the core axiom, a meta-theory that can interpret the axiom, a definition, and three essential theorems: non-vacuity, non-self-referential inconsistency, and maximality. In order to avoid any terminological circularity, the formulation of the core axiom that defines $\Omega$ is provided first:

Axiom of Maximal Cardinality (AMC): There exists a class cardinal $\Omega$ that is greater than any set cardinal in any theory T that can be consistently extended to a class theory proving the existence of $\Omega$.

Both T and the class theory that extends it can have a class-sized (and thus $\Omega$-sized) axiom class. Interpreting an axiom that is about theories and proving theorems about them is an endeavor that has similarities with proving Gödel’s ([8]) incompleteness theorems. With respect to the latter, we must distinguish the hierarchically organized countably recursive3 theories Gödel’s incompleteness theorems are about, from the meta-theoretical theory in which Gödel’s incompleteness theorems are proven. Gödel did not and could not treat this meta-theory as one of the countably recursive theories his theorems are about. Nevertheless, for AMC, we must assume the existence of a meta-theory M that interprets the notion ‘theory’ (including $\Omega$-sized theories) and not just the notion ‘countably recursive theory’.

M is granted the least restrictive bounds that $\Omega$ can render by being expressed in an infinitary logic $L_{\kappa ,\lambda }$ that extends the countable First-Order Logic and by having a signature $\tau$ with cardinality $\left|\tau \right|$ in which $\kappa =\lambda =\left|\tau \right|=\Omega$ (Karp [15], Barwise [1]). Here, $\kappa$ limits the size of conjunctions and disjunctions: only a number of sentences strictly smaller than $\kappa$ can be combined. $\lambda \le \kappa$ determines the maximum allowed length of a sequence of variables that can be bound by a single string of quantifiers (∃ or ∀). $\left|\tau \right|$ is the total number of the non-logical symbols (constants, functions, and predicates) that make up the signature $\tau$. This means that M can have $\Omega$-many axioms and that these can have any set ordinal length (strictly shorter than $\Omega$). AMC can be expressed in a shorter form thanks to the following definition:

Definition: A theory is Ω-consistent iff it can be consistently extended to a class theory proving the existence of $\Omega$.

A sentence is $\Omega$-consistent iff it is proven in an $\Omega$-consistent theory. A set/class is $\Omega$-consistent iff it provably exists in an $\Omega$-consistent set/class theory. The $\Omega$-consistency clause in AMC is required, because there are $\Omega$-inconsistent theories that prove the existence of an $\Omega$ that is smaller than some set cardinal. Then $\Omega$ would be greater than itself in M. Consequently, AMC is a fixed-point definition of $\Omega$.

Many investigations can be made to theories that prove AMC, have $\Omega$-many axioms, prove all the $\Omega$-consistent sentences, have proof-theoretic ordinal $\Omega$, and/or have the role of True Arithmetic for all class ordinals (Tarski et al. [28]). Also the role of an $\Omega$-long class ordinal Turing machine needs to be considered (Hamkins and Lewis [11], Koepke [16]). However, due to the maximality of $\Omega$, such investigations can lead to controversial results that violate standard logical reasoning. For this reason, they are beyond the scope of this paper. The focus is on AMC and $\Omega$. In the next section, three theorems are proven that counter three objections against AMC. The three theorems are:

• There exists a theory that is $\Omega$-consistent.
• AMC is not self-referentially inconsistent.
• $\Omega$ is a maximal cardinality.

3. Proving the Theorems

A natural objection against AMC is that it is vacuously true because there is no theory that is $\Omega$-consistent. Theorem 1 refutes this objection:

Theorem 1. 

There exists a theory that is Ω-consistent

Proof. 

Suppose toward a contradiction that there is no theory T that is $\Omega$-consistent: the set $x_T$ of the $\Omega$-consistent theories T is empty. This means that $\Omega$ can easily satisfy AMC. For example, ‘$\Omega ={\aleph }_0$’ satisfies AMC in this case, because ${\aleph }_0$ is then greater than any set cardinality in any theory T in $x_T$. But then there is a consistent theory T that is $\Omega$-consistent. Indeed, any consistent theory proving the existence of ${\aleph }_0$ is then $\Omega$-consistent for $\Omega ={\aleph }_0$, because it can be consistently extended to a class theory proving the existence of ${\aleph }_0$. That is a contradiction. Therefore, there exists an $\Omega$-consistent theory. □

Another natural objection against any definition of the absolute infinite is that this definition is self-referentially inconsistent. By making use of the consistency filter in AMC (only $\Omega$-consistent theories are considered), this objection is refuted by the following theorem:

Theorem 2. 

AMC is not self-referentially inconsistent

Proof. 

Suppose toward a contradiction that AMC is self-referentially inconsistent. Then there is no $\Omega$ that satisfies AMC. That means that for any $\Omega$ that is inserted in AMC, it holds that this $\Omega$ is not greater than some set cardinal $\kappa$ in some $\Omega$-consistent theory T in M. But then T proves the existence of an $\Omega$ that is not greater than some set cardinal in some theory that is $\Omega$-consistent, namely not greater than $\kappa$ in T itself. Therefore, T cannot be consistently extended to a class theory proving the existence of an $\Omega$ that satisfies AMC. This makes T $\Omega$-inconsistent. That is a contradiction. Therefore, AMC is not self-referentially inconsistent. □

At last, the third objection is that AMC – the axiom of maximal cardinality – does not have the meaning its name suggests, because $\Omega$ is not a maximal cardinality. Theorem 3 counters this objection:

Theorem 3. 

Ω is a maximal cardinality

Proof. 

Suppose toward a contradiction that $\Omega$ is not a maximal cardinality. This means that there must exist a class cardinal in M that is greater than $\Omega$. Consequently, $\Omega$ cannot be a proper class cardinal in M and must therefore be a set cardinal. As a result of that, there exists a consistent theory T in M in which $\Omega$ is a set cardinal. Because T is consistent and proves $\Omega$ exists, T can be consistently extended to a class theory proving the existence of $\Omega$. That makes T $\Omega$-consistent. But then AMC can be applied to T in M, which makes $\Omega$ greater than itself in M. That is a contradiction. Therefore, the size of $\Omega$ is maximal. □

4. Further Objections

In this section, it is argued that the provided definition of $\Omega$ is robust by answering several further objections to it.

4.1. $\Omega$ Succumbs to the Burali-Forti Paradox

The Burali-Forti paradox demonstrates that naively constructing the set of all set ordinals leads to a contradiction, namely that the constructed set is both an element of itself and not an element of itself. By introducing class ordinals in Section 2.1, $\Omega$ can be constructed as a proper class ordinal that is equal to the class of all set ordinals Ord_Ω, rather than the set of all set ordinals.

4.2. $\Omega$ Is Smaller Than $\Omega +1$

An ordinal $\Omega +1$ is not defined in any $\Omega$-consistent theory. (Without $\Omega$-consistency, $\Omega$ can be redefined to a set ordinal like 5 or $\omega$, making it trivially smaller than $\Omega +1$.) Since any successor construction requires $\{\Omega \}$, which is not defined, $\Omega +1$ cannot be formed. In a class theory, it is a category mistake to apply a successor operator to $\Omega$, because $\Omega$ is a proper class and not a set. If both a theory-specific ${\Omega }^{*}$ and ${\Omega }^{*}+1$ are well defined in the same consistent theory, then they are both set ordinals and therefore strictly smaller than $\Omega$. The same reasoning shows that, assuming $\Omega$-consistency, a meta-meta-theoretic level, a powerset operation, or any other enlarging operation cannot make $\Omega$ smaller than something else, because all these operations require the formation of $\{\Omega \}$.

4.3. $\Omega$ Should Not Occur Twice in AMC

$\Omega$ is referenced twice in AMC, just like self-referentially inconsistent definitions like “There exists a class cardinal $\Omega$ that is greater than the power set of $\Omega$.” However, being referenced twice does not by itself entail self-referential inconsistency, as acknowledged by Theorem 2. Moreover, the following axiom in which $\Omega$ occurs only once is self-referentially inconsistent: “There exists a class cardinal $\Omega$ that is greater than any set cardinal in any theory T that is consistent.” Among all the consistent theories T, there are T’s that define $\Omega$ as a set cardinal. A consistent theory that defines $\Omega$ as the smallest transfinite cardinal ${\aleph }_0$ is an example of such a T. Therefore, by referencing $\Omega$ twice in AMC, self-referential inconsistency is avoided instead of provoked.

4.4. $\Omega$ Comes in Many Varieties

An absolute infinite ${\Omega }_T$ is sometimes defined as the proper class ordinal or cardinal of a given countably recursive class theory T independent from AMC. In this case, there are many theory-dependent ${\Omega }_T$’s. $\Omega$ as defined via AMC is uniquely defined as greater than all the ${\Omega }_T$’s in any $\Omega$-consistent T of this kind. This is a unique, theory-independent definition.

5. Conclusions

This paper shows first that a theory-dependent maximality can be defined via proper class ordinals and proper class cardinals. For a theory-independent maximality, it uses a fixed-point axiom of maximal cardinality AMC that asserts the existence of a cardinality $\Omega$ that transcends all set cardinals in all $\Omega$-consistent theories. This counters height potentialism. Interpreting AMC requires a theory that interprets the notion ‘theory’.

Three essential objections can be made: (1) that AMC is vacuously true because there is no theory that is $\Omega$-consistent, (2) that AMC is self-referentially inconsistent, and (3) that $\Omega$ is not a maximal cardinality. These objections are countered by three theorems that are proven. Some further objections are also rebutted: that $\Omega$ succumbs to the Burali-Forti paradox, that $\Omega$ is smaller than $\Omega +1$, that $\Omega$ should not occur twice in AMC, and that $\Omega$ comes in many varieties.

Height potentialism will not be entirely abandoned by this absolutist philosophy. It remains viable for those who reject AMC or theories that interpret the notion ‘theory’. After all, even the axiom of infinity is not universally accepted, in particular not by finitists and philosophers who interpret a potential infinity as being about the least upper bound of the natural numbers.4 The aim of this paper is to show that an absolute infinite is not incoherent and can be consistently axiomatized, making it a viable philosophical phenomenon.