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 [27], Blondé [3]), but can be used for various other purposes too: in metaphysics and modal realism to describe the size of Lewis’ ([19]) logical space or the plenitude (Blondé [2]), in computer science to have the ultimate oracle (Burgin [6]) or ordinal2 machine (Koepke [17]), in epistemology to formulate omniscience (Heylen [15]), and in the philosophy of mathematics to have a maximal entity (Sutto [28]).

Cantor’s idea continues to provoke controversy. Welch and Horsten ([32]) review Cantor’s conception of the set-theoretic universe as a completed infinity and prefer it above Zermelo’s ([34]) conception, because Cantor’s universe includes the modern large cardinals via reflection principles. Livadas ([22]) also discusses Cantor’s absolute infinite in light of the modern large cardinals and argues that it is proof-theoretically unattainable. Gutschmidt and Carl ([11]) 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 [35], Putnam [25], Parsons [24], Hamkins [13], Linnebo [20], Hellman and Shapiro [14], Linnebo and Shapiro [21], Brauer et al. [4], Sutto [28]). 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 [13], Scambler [26], Meadows [23], Gorbow [10]). 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 axiom of maximal strength (AMS) that is consistent with a maximally strong meta-theory $T_M$, establishes the existence of an absolutely maximally large infinity $\Omega$. The next section introduces $\Omega$ via class ordinals, class cardinals, the axiom AMS, some definitions, the meta-theory $T_M$, and eight theorems. In Section 3, the theorems are proven on the basis of AMS and $T_M$. After that, in Section 4, a range of objections to $\Omega$ and its existence 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 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 maximal ordinals and cardinals, namely the proper class ordinal and the proper class cardinal, and unambiguous terms to refer to set ordinals and set cardinals. A set $\alpha$ is a set ordinal if and only if (henceforth iff) (Kunen [18], p. 16):

1.

$\alpha$ is a transitive set.

2.

($\alpha$, ∈) is a well-ordering.

The Burali-Forti ([5]) paradox shows that the collection of all set ordinals Ord cannot be a set (Kunen [18], p. 17). If such a collection would be a set, it would be greater than 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 MK (Morse–Kelley) can be used when a theory about sets and classes is needed (Wang [31]). With this notion of classes, class ordinals can be defined as an extension of set ordinals. A class C is a class ordinal iff:

1.

C is a transitive class.

2.

(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 an MK-class-level generalization of the von Neumann definition of (set) ordinal. By the usual ordinal analysis (Kunen [18], 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 ([18], p. 17), the intersection of two class ordinals is again a class ordinal. In MK, 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 ([18], 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 within any given model of MK. Ord is – again, within any given model – also the supremum of all set ordinals and it is maximal: no class ordinal in the model is greater than 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 [18], 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.

By using the axiom of choice, every set cardinal can be equated with its initial set ordinal and the proper class cardinal $\left|V\right|$ with the proper class ordinal Ord. In this way, we also have a proper class cardinal that is maximal and unique. Here, $\left|V\right|$ is the size of the von Neumann universe of all sets V. Nevertheless, as we will see in what follows, there are infinitely many non-maximally strong theories that have infinitely many non-maximally large models, and $\left|V\right|$ and Ord are different in each of these. The symbol $\Omega$ is reserved for the $\left|V\right|$ and Ord that provably exist according to the maximally strong theory $T_M$ (to be defined later).

2.2. The Absolute Infinite Through Familiar Concepts

In this paper, the absolute infinite is viewed via the Axiom of Maximal Strength (AMS). Nevertheless, there are ways to conceive of the absolute infinite through concepts that are already familiar. To begin with, AMS is a more advanced version of the following finite second-order Axiom of Maximal Cardinality (AMC): “There exists a class cardinal with the size of the class that contains every set.” AMC is second-order because it quantifies over all first-order definable objects, namely the sets.

According to a naive, theory-unaware view, the class cardinal in AMC is $\Omega$ and the class that contains every set is the true universe V. However, axiomatic set theory has much progressed since its early days, such that there are now many mutually inconsistent and relatively consistent theories that interpret AMC and its proper classes differently. Each theory has infinitely many universe models that make all the axioms of the theory true. For example, a model of a theory in which CH is true is irreconcilable with the models of a theory in which ¬CH is true. Relatively consistent theories do not seem to make this situation better: a universe model of one theory can then be interpreted to be a set that is provably smaller than a universe model of the other theory.

This state of affairs thus results in infinitely many different false points of view on the true V and its cardinality $\Omega$, which is why the existential status of the true V, and certainly that of $\Omega$, is currently undermined. This paper aims to rehabilitate $\Omega$ and the true V by pointing out that they can be correctly viewed through the lens of a maximally strong theory. The theories mathematicians use for practical purposes are necessarily insufficient to prove the existence of $\Omega$ and the true V.

The solution to this problem is uncomplicated: just like AMC is a finite second-order axiom that quantifies over all first-order-set-definable set objects, AMS is a finite second-order axiom that quantifies over all first-order-set-definable set theories. In both cases, transfinitely long definitions are explicitly taken into consideration. This implies that the concept `theory’ is itself elevated to a more theoretical level. Theories with transfinitely long definitions may not be practically manageable, however, this problem is not a good argument to claim that $\Omega$ is poorly defined.

The counterargument that AMC does not suffice to prove the existence of $\Omega$ uses the exact same theoretical approach by making a second-order quantification over theories: “No recursively enumerable, nor even any first-order-set-interpretable theory, can prove the existence of a maximally large class cardinal $\Omega$.” Without such a second-order quantification, the counterargument stalls. The claim that some particular recursively enumerable theory like MK cannot prove the existence of $\Omega$ may be true, it remains without consequences by the fact that some strong extension of MK might very well prove the existence of $\Omega$.

Another example of a second-order quantification over theories, namely all recursively enumerable theories, is Gödel’s ([9]) incompleteness theorems. The first can be formulated as: “No recursively enumerable theory is capable of proving all truths about the arithmetic of natural numbers.” Consequently, the argument that AMS is somehow false or illegitimate because it makes a second-order quantification over theories cannot be maintained.

A second way to conceive of the absolute infinite through a concept that is already known is through the concept $L_{\infty ,\infty }$ (Karp [16], Barwise [1]), which is an infinitary first-order language that permits set-many conjunctions and disjunctions over set-many variables.3 Because every set is well-founded in MK, it is always possible to write down a parameter-free set-ordinal-length formula that describes all the elements in its transitive closure. That makes every set in the true universe of all sets definable by a single sentence in $L_{\infty ,\infty }$. In the other direction, every sentence consists of symbols and can be mapped to a unique set. This results in the following Axiom of Maximal Expressivity (AME): “There exists a class cardinal that is equal to number of sentences in $L_{\infty ,\infty }$.” This class cardinal in AME is thus equal to the absolute infinite $\Omega$ of AMC. The language $L_{\infty ,\infty }$ will also be used later on to construct a maximally powerful infinitary first-order logic and a maximally strong infinitary theory.

2.3. Is Absolutely Everything a Class?

Choosing for a class theory such as MK in an absolutist context comes with the ontological commitment that absolutely every object is a class. The defense of this commitment is that all logically possible things – including classes – are objects that contain information and that sufficiently large information objects can be decoded without any external code book. Indeed, they can use an arbitrarily large internal code book. In this way, objects in all ontological categories can be a class: sets, collections, numbers, graphs, Ordinal Turing Machines (Hamkins and Lewis [12], Koepke [17]), axiomatic theories, models of axiomatic theories, and even physical worlds. There is no experiment that an intelligent physical being can conduct to find out whether or not it is a class or is computed by an abstract Ordinal Turing Machine. This is purely a matter of convention.

A consequence of this approach is that we get a multi-layered ontology in which the same object can have contradictory properties indexed by different layers. A collection can then be a well-founded class in the deepest layer, while being non-well-founded in a higher layer. A physical replica of a class with five members can have the property of being an enormous class containing the information of all the sub-atomic particles that make up the replica in the deepest layer, while being a class with five members in the highest layer. Likewise, a data structure in a software program can be a two-valued string of bits in the deepest software layer, while being a string of characters or a photo in the highest layer. When we say that absolutely everything is a class, we refer to the deepest ontological layer.

2.4. Axiom, Definitions, Meta-Logic, and Meta-Theory

This section provides the axiom AMS, some definitions, a meta-logic called ${\Omega }^{+}$-logic,4 and a meta-theory $T_M$ that is ${\Omega }^{+}$-consistent with AMS. Since AMS is about theories, ZFC (Zermelo–Fraenkel set theory with the axiom of choice, Zermelo [34]) and MK are chosen as a foundation for set theories and class theories, respectively. We begin with AMS:

Axiom of Maximal Strength (AMS): There exists a class cardinal that does not ${\Omega }^{+}$-provably exist in any MK-extending $T_M$-fragment that has an injective theorem-preserving ${\Omega }^{+}$-interpretation into a ZFC-extending $T_M$-fragment that is definable by a formula without parameters in the infinitary first-order language $L_{\infty ,\infty }$, where ${\Omega }^{+}$-provability and ${\Omega }^{+}$-interpretability are defined in a logic that can make a number of deductive steps equal to the number of sentences in $L_{\infty ,\infty }$.

In this way, the definition of $\Omega$ depends on that of $L_{\infty ,\infty }$. The cardinal mentioned will provisionally be called AMS’s class cardinal. It will be proven to be equal to the class ordinal ${Ord}_{T_M}$ in Theorem 5 and equal to the absolute infinite $\Omega$ in Theorem 8. The following definitions are needed to interpret AMS:

Definition: A theory T is a $T_M$-fragment iff it is an infinitary fragment of the meta-theory $T_M$ (hence $T\subseteq T_M$).

Note that also the meta-theory $T_M$ is a $T_M$-fragment and that finitary theories are also considered to be infinitary theories in this paper.5

Definition:${\Omega }^{+}$-logic is the infinitary first-order logic with identity and membership that can make a number of deductive steps that is equal to the number of sentences that can be expressed in the infinitary first-order language $L_{\infty ,\infty }$.

Definition:${\Omega }^{+}$-provability and ${\Omega }^{+}$-interpretability mean provability and interpretability using ${\Omega }^{+}$-logic.

Proofs in ${\Omega }^{+}$-logic can be proper-class-sized, rather than being set-sized or finite. If proofs must remain set-sized, they can, for example, never witness that a given structure is isomorphic to the true universe of all sets V.

Definition: An infinitary theory T is ${\Omega }^{+}$-consistent iff no contradiction can be derived from the axioms of T, using ${\Omega }^{+}$-logic.

These definitions use the convention that a ${\kappa }^{+}$-logic can make at most $\kappa$ deductive steps and a $\kappa$-logic strictly less than $\kappa$ steps, with $\kappa$ some class ordinal. Ordinary (finitary) consistency is then called $\omega$-consistency, and the $\omega$-consistency Gödel used in his incompleteness theorem is then ${\omega }^{+}$-consistency.

In fact, ${\Omega }^{+}$-consistency genuinely deserves to be called consistency, given that other types of consistency are entailed by ${\Omega }^{+}$-consistency. However, set-limited deductive systems may fail to derive a contradiction in a theory that is not even $\Omega$-consistent and thus conclude that the theory is consistent. Consequently, there are `consistent’ theories that do not have a model. Furthermore, if arbitrary $L_{\infty ,\infty }$-sentences are allowed, while the deduction of neither ${\Omega }^{+}$-logic nor even $\Omega$-logic6 is used in the definition of consistency, one comes to the conclusion that infinitary theories are not well behaved with respect to the principle that every consistent theory has a model (Dickmann [8]). The following definitions enable AMS to be expressed in a shorter form:

Definition: A ZFC-extending $T_M$-fragment T is first-order-set-definable iff T is definable by a formula without parameters in the infinitary first-order language $L_{\infty ,\infty }$.

This means that the theorems of T and the sentences that assert the existence of the set cardinality of the set of theorems of T and the supremum of the lengths of the theorems of T, are all theorems that are provable by a pure (first-order-set-definable) infinitary ZFC extension.

Definition: An MK-extending $T_M$-fragment is first-order-set-interpretable iff it has an injective theorem-preserving ${\Omega }^{+}$-interpretation into a first-order-set-definable ZFC-extending $T_M$-fragment.

First-order-set-interpretability can also be applied to classes:

Definition: A class is first-order-set-interpretable iff it provably exists in a first-order-set-interpretable MK-extending $T_M$-fragment.

Otherwise, the class is non-first-order-set-interpretable. Combining the definitions results in the following shortened expression of AMS:

Shortened AMS: There exists a non-first-order-set-interpretable class cardinal.

Some more definitions are needed to efficiently express the presuppositions about $T_M$ and the theorems in the next section:

Definition: A class x is T-provable iff the theory T proves that x exists.

Definition: An infinitary theory T has a model iff there exists a model that makes all the axioms of T true.

Definition: An infinitary theory T is ${\Omega }^{+}$-sound iff no falsehoods can be derived from the axioms of T, using ${\Omega }^{+}$-logic.

Definition: The infinitary first-order language $L_{{\infty }^{+},{\infty }^{+}}$ permits a number $\kappa$ of conjunctions and disjunctions over a number $\lambda$ of variables, where $\kappa$ can be any arbitrarily large class cardinal and $\lambda \le \kappa$.

While $L_{\infty ,\infty }$ has proper-class-many set-ordinal-long sentences, $L_{{\infty }^{+},{\infty }^{+}}$ has proper-class-many class-ordinal-long sentences.

Definition: An MK-extending $T_M$-fragment T is ${\Omega }^{+}$-complete iff for every sentence $\varphi$ in the language $L_{{\infty }^{+},{\infty }^{+}}$, either $T⊢\varphi$ or $T⊢\neg \varphi$.

Definition:${Ord}_T$ of an MK-extending $T_M$-fragment T is the class of all T-provable set ordinals.

Definition: A theory T is maximally strong iff in any ${\Omega }^{+}$-sound theory $T_S$, all $T_S$-provable objects are T-provable objects.

The proofs of the theorems require a meta-theory $T_M$ about which we need to make some presuppositions:

1.

$T_M$ is ${\Omega }^{+}$-sound.

2.

$T_M$ is ${\Omega }^{+}$-complete.

3.

$T_M$ is an infinitary extension of MK.

Presupposition 1: This presupposition entails ${\Omega }^{+}$-consistency. If $T_M$ were ${\Omega }^{+}$-inconsistent, $T_M$ + AMS would be ${\Omega }^{+}$-inconsistent, in which case ${\Omega }^{+}$-logic derives a contradiction from the assertion that $\Omega$ exists. This amounts to the conclusion that $\Omega$ does not exist according to $T_M$. For the adherents of width potentialism and truth relativism, presupposing soundness is not a standard practice, because that fixes the truth value of every axiom of the theory in a way that makes all the theories unsound that contradict these axioms. However, from a theory that is not ${\Omega }^{+}$-sound, ${\Omega }^{+}$-logic derives theorems that are false. Because of this argument, ${\Omega }^{+}$-soundness is an important requirement in an absolutist orientation. Even if $T_M$ proves its own ${\Omega }^{+}$-soundness, provided it is ${\Omega }^{+}$-sound, $T_M$ will also prove its own ${\Omega }^{+}$-soundness, provided it is not ${\Omega }^{+}$-sound. This makes the assertion that $T_M$ is ${\Omega }^{+}$-sound necessary as a presupposition.

Presupposition 2: This presupposition is needed to exclude that $T_M$ has a meta-theory that proves theorems $T_M$ cannot prove. Having such a meta-theory would make $T_M$ objectively inferior as a theory and inappropriate as a basis to prove truths about all of reality.

Presupposition 3: MK is adopted because it is an extension of ZFC, the standard axiomatic foundation for mathematics, that can accommodate AMS’s class cardinal. Because of our mathematical experience with this foundation, MK can be presupposed to be sound and ${\Omega }^{+}$-sound, as entailed by Presupposition 1. If someday a consensus grows that MK or ZFC is unsound, it needs to be replaced with a foundation that is sound. While NBG (von Neumann–Bernays–Gödel) (von Neumann [30]) is a conservative extension of ZFC with respect to set-theoretic sentences, MK extends NBG with a full truth predicate for first-order set-theoretic formulas. Because of this, $T_M$ can express the `first-order-set-definable’ in AMS as a legitimate predicate. Furthermore, because NBG cannot prove the existence of impredicatively defined classes, $T_M$ could not be maximally strong if it did not extend MK.

With these definitions and presuppositions, the following theorems are proven in Section 3:

1.

An infinitary first-order theory has a model iff it is ${\Omega }^{+}$-consistent.

2.

Every $T_M$-fragment has a model.

3.

A $T_M$-fragment cannot be both ${\Omega }^{+}$-complete and first-order-set-interpretable.

4.

An MK-extending $T_M$-fragment T is first-order-set-interpretable if ${Ord}_T$ is first-order-set-interpretable.

5.

${Ord}_{T_M}$ is a non-first-order-set-interpretable class cardinal.

6.

$T_M$ is ${\Omega }^{+}$-consistent with AMS.

7.

$T_M$ is maximally strong.

8.

${Ord}_{T_M}$ is an ${\Omega }^{+}$-consistent version of Cantor’s absolute infinite $\Omega$.

3. Proving the Theorems

In this section, the theorems are proven that work toward the conclusion that AMS’s non-first-order-set-interpretable class cardinal can be equated with an ${\Omega }^{+}$-consistent version of Cantor’s absolute infinite $\Omega$. In order to prove the ${\Omega }^{+}$-consistency of AMS relative to $T_M$ in theorem 6, some preparatory theorems are needed first:

Theorem 1. 

An infinitary first-order theory has a model iff it is ${\Omega }^{+}$-consistent.

Proof. 

Because any model that makes all the axioms of an infinitary theory true counts as a model of the theory, an infinitary first-order theory has a model iff no contradictions can be derived from the axioms of the theory using a first-order logic that can derive all the deductive consequences of the axioms of the theory. A first-order logic can derive all the deductive consequences of the axioms of an infinitary first-order theory iff the number of deductive steps the logic can make is greater or equal to the total number of conjunctions and disjunctions in the conjunction of all the axioms of the theory. ${\Omega }^{+}$-logic is a first-order logic that can make a number of deductive steps that is equal to the number of $L_{\infty ,\infty }$-sentences. The number of sentences in $L_{\infty ,\infty }$ is proper-class-sized and is therefore equal to the total number of conjunctions and disjunctions in the conjunction of all the sentences in $L_{{\infty }^{+},{\infty }^{+}}$. Every infinitary first-order theory can be expressed by a single sentence in $L_{{\infty }^{+},{\infty }^{+}}$. As a result, all the deductive consequences of any infinitary first-order theory can be derived by ${\Omega }^{+}$-logic. Consequently, an infinitary first-order theory has a model iff no contradictions can be derived from the theory using ${\Omega }^{+}$-logic. Therefore, an infinitary first-order theory has a model iff it is ${\Omega }^{+}$-consistent. □

Theorem 2. 

Every $T_M$-fragment has a model.

Proof. 

Let T be a $T_M$-fragment: $T\subseteq T_M$. By Presupposition 1 ($T_M$ is ${\Omega }^{+}$-sound), no falsehoods can be derived from the axioms of $T_M$ using ${\Omega }^{+}$-logic, and thus no contradictions either. Therefore, $T_M$ is ${\Omega }^{+}$-consistent. Because T is a fragment of $T_M$, T is also ${\Omega }^{+}$-consistent. The fragments of $T_M$ are expressible as single-sorted first-order theories in the language $L_{{\infty }^{+},{\infty }^{+}}$, where the domain of discourse consists of classes. Consequently, T qualifies as an infinitary first-order theory. By Theorem 1, T then has a model. □

Theorem 3. 

A $T_M$-fragment cannot be both ${\Omega }^{+}$-complete and first-order-set-interpretable.

Proof. 

Suppose toward a contradiction that T is a $T_M$-fragment that is both ${\Omega }^{+}$-complete and first-order-set-interpretable. Since T is ${\Omega }^{+}$-complete, for every sentence $\varphi \in L_{{\infty }^{+},{\infty }^{+}}$, either $T⊢\varphi$ or $T⊢\neg \varphi$. Hence T has at least as many distinct theorems as there are distinct $L_{\infty ,\infty }$-sentences. Since $L_{\infty ,\infty }$ contains sentences of arbitrarily large set-ordinal lengths for $T_M$-provable set ordinals, the class of its sentences is a proper class in $T_M$. Consequently, Thm(T), the class of all the theorems of T, is also a proper class in $T_M$.

Because T is first-order-set-interpretable, there exists an injective theorem-preserving ${\Omega }^{+}$-interpretation $\tau :T\to T_{ZFC}$ of T into a first-order-set-definable ZFC-extending $T_M$-fragment $T_{ZFC}$. Since $\tau$ is injective and theorem-preserving, distinct theorems of T yield distinct theorems of $T_{ZFC}$. Thus the collection of all theorems of $T_{ZFC}$ must also be a proper class in $T_M$. But $T_{ZFC}$ is first-order-set-definable, which means that its collection of theorems has set cardinality in $T_M$. That is a contradiction. Therefore, a $T_M$-fragment cannot be both ${\Omega }^{+}$-complete and first-order-set-interpretable. □

Theorem 4. 

An MK-extending $T_M$-fragment T is first-order-set-interpretable if ${Ord}_T$ is first-order-set-interpretable.

Proof. 

Suppose T is an MK-extending $T_M$-fragment and ${\mathrm{Ord}}_T$, the class of all the T-provable set ordinals, is first-order-set-interpretable. Then ${Ord}_T$ provably exists in an MK-extending $T_M$-fragment $T_2$ that has an injective, theorem-preserving ${\Omega }^{+}$-interpretation $\tau$ in a first-order-set-definable ZFC-extending $T_M$-fragment $T_3$. Because T and $T_2$ are $T_M$-fragments, they have a model according to Theorem 2. Moreover, because ${Ord}_T$ provably exists within the infinitary MK-extension $T_2$, $T_2$ can internally construct a model of T through impredicative class comprehension and class transfinite recursion (recursive constructions of classes along well-orderings) in MK. By impredicative class comprehension, $T_2$ can prove that its internal construction satisfies the infinitary axioms of T. By class transfinite recursion, $T_2$ can iterate the model-building steps (like taking power sets and definable subsets) unboundedly up to ${Ord}_T$. This inner-model construction warrants that there exists an injective, theorem-preserving ${\Omega }^{+}$-interpretation ${\tau }_1:T\to T_2$. By composing ${\tau }_1$ and $\tau$, we get ${\tau }_3=\tau \circ {\tau }_1:T\to T_3$. ${\tau }_3$ is injective because the composition of two injective ${\Omega }^{+}$-interpretations is injective. ${\tau }_3$ is also theorem-preserving: $T⊢\varphi$ iff $T_2⊢{\tau }_1\left(\varphi \right)$ iff $T_3⊢\tau \left({\tau }_1\left(\varphi \right)\right)$. Consequently, T has an injective, theorem-preserving ${\Omega }^{+}$-interpretation in a first-order-set-definable ZFC-extending $T_M$-fragment, namely $T_3$. Therefore, T is first-order-set-interpretable. □

Theorem 5. 

${Ord}_{T_M}$ is a non-first-order-set-interpretable class cardinal.

Proof. 

According to Theorem 3, the $T_M$-fragment $T_M$ cannot be both ${\Omega }^{+}$-complete and first-order-set-interpretable. By Presupposition 2, $T_M$ is ${\Omega }^{+}$-complete. Therefore, $T_M$ is not first-order-set-interpretable. According to Theorem 4, the MK-extending $T_M$-fragment $T_M$ is first-order-set-interpretable if ${Ord}_{T_M}$ is first-order-set-interpretable. By contrapositive, ${Ord}_{T_M}$ is not first-order-set-interpretable. By the axiom of choice, ${Ord}_{T_M}$ is a class cardinal by being the first class ordinal with size |${Ord}_{T_M}|$. Therefore, ${Ord}_{T_M}$ is a non-first-order-set-interpretable class cardinal. □

Theorem 6. 

$T_M$ is ${\Omega }^{+}$-consistent with AMS.

Proof. 

According to Theorem 5, ${Ord}_{T_M}$ is a non-first-order-set-interpretable class cardinal. Therefore ${Ord}_{T_M}$ satisfies the existential assertion of AMS (There exists a non-first-order-set-interpretable class cardinal). Because $T_M$ extends MK (Presupposition 3), we also have that $T_M$ proves ${Ord}_{T_M}$ to exist. Since $T_M$ is also ${\Omega }^{+}$-consistent by the ${\Omega }^{+}$-soundness of $T_M$ (Presupposition 1), $T_M$ is ${\Omega }^{+}$-consistent with AMS. □

Theorem 7. 

$T_M$ is maximally strong.

Proof. 

Let $T_S$ be any arbitrary theory that is ${\Omega }^{+}$-sound. Assume $T_S$ proves the existence of an object x. In the language $L_{{\infty }^{+},{\infty }^{+}}$, x corresponds to a theorem that can be written as $T_S⊢\exists x\varphi \left(x\right)$. Because $T_S$ is ${\Omega }^{+}$-sound, the sentence $\exists x\varphi \left(x\right)$ must be true. By Presupposition 2 ($T_M$ is ${\Omega }^{+}$-complete) $T_M$ either proves this sentence or proves its negation. By Presupposition 1, ($T_M$ is ${\Omega }^{+}$-sound) $T_M$ cannot prove the negation of this sentence, because that negation is false. Therefore, $T_M⊢\exists x\varphi \left(x\right)$. Since this argument applies to any object proven to exist by any ${\Omega }^{+}$-sound theory $T_S$, all $T_S$-provable objects are $T_M$-provable objects. By definition, this makes $T_M$ maximally strong. □

Theorem 8. 

${Ord}_{T_M}$ is an ${\Omega }^{+}$-consistent version of Cantor’s absolute infinite Ω.

Proof. 

According to Theorem 7, $T_M$ is maximally strong. This means that for any theory T to prove the existence of an object that is not a $T_M$-provable object, T must prove falsehoods. Consequently, $T_M$ is the only legitimate source of truth about reality and about the question whether ${Ord}_{T_M}$ is an ${\Omega }^{+}$-consistent version of Cantor’s absolute infinite. Of all the objects that $T_M$ proves to exist, none has a cardinality that is greater than the cardinality of ${Ord}_{T_M}$. By the axiom of choice, ${Ord}_{T_M}$ is both a class cardinal and a class ordinal. Furthermore, as a result of the ${\Omega }^{+}$-soundness of $T_M$ (Presupposition 1), $T_M$ is an ${\Omega }^{+}$-consistent theory. Therefore, ${Ord}_{T_M}$ is an ${\Omega }^{+}$-consistent version of Cantor’s absolute infinite $\Omega$. □

4. Objections

In this section, it is argued that the provided definition of $\Omega$ is robust by answering several 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 in $T_M$, 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 theory that is ${\Omega }^{+}$-consistent with $T_M$. (Without this 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 $\left\{\Omega \right\}$, 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 $T_M$-consistent theory, then they are both set ordinals and therefore strictly smaller than $\Omega$. The same reasoning shows that, assuming $T_M$-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 $\left\{\Omega \right\}$.

The usage of the `+’ symbol in ${\Omega }^{+}$-logic may seem to suggest that something larger than $\Omega$ exists. However, it indicates that proofs can have length $\Omega$. The notation without the `+’ symbol is already needed to indicate that proofs can have any arbitrary set-ordinal length, but not length $\Omega$.

4.3. $\Omega$ and the Universe V are Always Extensible

A possible objection of the height potentialist is that the universe of all sets V is always extensible. However, `universe’ is not a good name for something that is extensible. The terminologically ideal solution is that V is a non-extensible entity, just like its cardinality $\Omega$. If such a V cannot be defined via the standard axiomatic toolbox, then this is a sign that this toolbox needs to be enriched and not that a universe is an extensible entity. A non-maximal model of an incomplete theory is rather a `universe view’ than the true universe itself.

4.4. $\Omega$ Is Already Used in AMS to Define $\Omega$

The objection could be made that AMS uses ${\Omega }^{+}$-provability, which requires a definition of ${\Omega }^{+}$-logic, and that the latter definition requires a definition of $\Omega$. Therefore, the definition of $\Omega$ is circular. However, even though ${\Omega }^{+}$-logic uses the symbol $\Omega$ in its name, its definition is derived from the definition of $L_{\infty ,\infty }$, which is the language that contains every set-long infinitary first-order sentence. Therefore, just like AMC and AME, AMS does not require prior knowledge about $\Omega$.

4.5. $T_M$ Violates Gödelian Limitations

If the meta-theory $T_M$ were recursively enumerable, then $T_M$ would have been subjected to the limitations of Gödel’s incompleteness theorems. However, a recursively enumerable theory can prove only a countable number of theorems, while $T_M$ proves an absolutely infinite number of theorems, namely one for each sentence in $L_{{\infty }^{+},{\infty }^{+}}$, as a result of Presupposition 2. $T_M$ is therefore two steps beyond a recursively enumerable theory (with non-recursively enumerable first-order-set-interpretable theories as intermediate step) such that Gödel’s incompleteness theorems cannot be applied.

5. Conclusions

This paper shows first that a theory- and model-dependent maximality can be defined via proper class ordinals and proper class cardinals. For a maximality that is absolute, it uses the Axiom of Maximal Strength AMS that asserts that there exists a class cardinal $\Omega$ that does not ${\Omega }^{+}$-provably exist in any MK-extending $T_M$-fragment that has an injective theorem-preserving ${\Omega }^{+}$-interpretation into a ZFC-extending $T_M$-fragment that is definable by a formula without parameters in the infinitary first-order language $L_{\infty ,\infty }$. ${\Omega }^{+}$-provability and ${\Omega }^{+}$-interpretability are defined using ${\Omega }^{+}$-logic, which is powerful enough to enable as many deductive steps as the number of sentences in $L_{\infty ,\infty }$.

AMS is a more advanced version of the Axiom of Maximal Cardinality (AMC): “There exists a class cardinal $\Omega$ with the size of the class that contains every set.” However, when AMC is analyzed in the context of multiple recursively enumerable – or otherwise non-maximally strong – class theories, multiple non-maximal $\Omega$’s appear. The solution is the following: just like AMC is a finite second-order axiom that quantifies over all first-order-set-definable set objects, AMS is a finite second-order axiom that quantifies over all first-order-set-definable set theories. It can then be proven that AMS is ${\Omega }^{+}$-consistent with an ${\Omega }^{+}$-complete meta-theory $T_M$ and that ${Ord}_{T_M}$ equals $\Omega$.

Potentialism in set theory will not be entirely abandoned by this absolutist philosophy. It remains viable for those who reject AMS or second-order quantification over theories. 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.7 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.