The Ontological Import of Adding Proper Classes

In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenﬁeld have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of the same formula. We argue that the ﬁrst makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. More-over, we will modernize Shoenﬁelds proof, emphasizing its relation to Herbrands theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are powerful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological conten

Within the context of a formal theory, an assertion is ontologically committing if it expresses a closure property of the intended models. For example, the power set and the union axioms express closure properties in set theories.
If X and Y are elements in our set theory model, then the axioms guarantee that their union and ther power sets are also in the model. This property represents how an axiom generates ontological import. A detailed analysis of this conception leads to an understanding of existential aspects in formal theories (see [4]). But it is of no help if we want to compare different theories regarding ontology. We propose a different approach for this kind of comparison: If T 1 and T 2 are formal theories and T 2 can reduce T 1 but T 1 cannot reduce T 2 using a natural method of ontological reduction given in a metatheory, then the ontological content of T 2 is greater than that of T 1 . Now, what is a natural method of ontological reduction? We expect that if T 1 reduces T 2 , then the consistency of T 1 can be proved from the consistency of T 2 in the corresponding metatheory. This and other related questions are dealt with in [1], which is the source of the main results in this paper. Nevertheless, we will not be concerned with this question in its full generality here, as we will concentrate on methods of interpretations between first-order theories in a finitary metatheory. More precisely, using the method of interpretations and the above conception of ontological comparison, we prove that the ontological content of NBG is greater than the ontological content of ZF. This result goes against the received view according to which those theories are equivalent.
We will analyze the received view and point out its insufficiencies. It is based on the folklore conservativity result: NBG is conservative with respect to ZF. A detailed finitary proof of this result will be provided, as there is basically no modern proof for this in the literature. We will argue that the ontological reduction operating here is not conclusive, for the corresponding reduction method is partial. Hence, it is not clear if it gives a right transposition of the ontology. Also, it is always possible to strengthen the metatheory and weaken the reduction method in order to trivialize the comparison.
Therefore, for the equivalence claim, it is not enough to have one mutual comparison according to some reduction method in some metatheory. In the opposite direction, it is very significant to have an asymmetric comparison by a standard method of reduction, and this result will be proved subsequently, together with other related results. The absence of interpretation from NBG to ZF thus understood is a strong evidence for the thesis that the ontological content of the former surpasses that of the latter. The above reduction gives, in particular, a model-theoretic proof that if ZF is consistent, so is NBG. However, a bolder philosophical conclusion from this is that the ontological content of NBG is already present in ZF, as it can be easily fulfilled by the definable classes lurking in models of ZF. We claim that the bolder conclusion is unwarranted. The problem is that the metatheory in which the reduction takes place is too strong and the difference may be obliterated by its excessive strength. To make the point clear, assume that we were interested in comparing theories T 1 and T 2 in a metatheory which happens to be strong enough to prove the consistency of both theories. The equivalence between the consistency of T 1 and that of T 2 is, therefore, valid in such a metatheory, but no ontological comparison can be drawn.
From a proof of equiconsistency in some metatheory, one cannot conclude ontological equivalence. However, there is another reduction of NBG to ZF providing a finitary equiconsistency result due to Shoenfield. Although the above argument does not apply to Shoenfield's reduction, which takes place in a finitary metatheory, we still claim that the bolder conclusion that the ontological content of NBG is already present in ZF is unwarranted. The problem is within the reduction method itself: It does not provide a reduction of the quantification over class variables, which gives NBG its extra ontological content, but rather just shows that the quantificational reasoning with class variables in NBG is dispensable and can be avoided in proofs of ZF-sentences. Therefore, there is no real reduction taking place here, there is no transposition of ontology, and the ontological equivalence does not follow.
Shoenfield's proof will be given in the next section and meticulously analyzed to support our claim.

Shoenfield's finitary reduction
The finitary proof of equiconsistency between NBG and ZF was provided by Shoenfield in the article "A relative consistency proof" ( [9]). This article is written in the language of Principia Mathematica by Whitehead and Russell, and makes extensive use of the techniques developed in Grundlagen der Mathematik by Hilbert and Bernays. For this reason, we have developed this section by an excavation, a reverse engineering, in which the tools used were unraveled by the clues left in the article. In addition, changing the axiomatic system (see [7]) poses several additional difficulties and, in many instances, the proof changes significantly.
We will expose here the technique used for the equiconsistency proof. Before doing that, however, we need to remember the finitary proof technique of Herbrand's theorem. For this, we will go through the necessary definitions, then we will enunciate Hilbert-Ackermann's theorem and, finally, the necessary part of the strategy for establishing Herbrand's theorem.  The finitary proof of this theorem can be found in [5, p. 48 -52]. It is important to bear in mind this theorem since it is equivalent to the existential case of Herbrand's theorem.
Definition 5 Let Q be a quantifier ∀ or ∃, then a prenex formula is of the form: being θ a open formula.
We can write the prenex form, without loss in generality, with explicit quantifiers ∀ e ∃, instead of using the Q. The general formula is of the form: being x the sequence of free variables in the matrix θ. Using this notation will simplify the proof, since each quantifier ∀ and ∃ is treated differently.
Definition 6 A formula α is existential when α is a prenex formula of the form ∃x 1 ∃x 2 . . . ∃x n θ, being θ an open formula.
From prenex formulas, we have the following theorem: Theorem 7 For any formula α, there is a α ′ , such that: 3. α ′ is obtained by a primitive recursive procedure; We call α ′ the prenex form of α.
Next we will expose Herbrand's normal form. It can be understood as the representation of any formula by an existential formula through a procedure of elimination of universal quantifiers. Such elimination is due to the introduction of function symbols in language.
Definition 8 (Herbrand's normal form) Let α be any formula, we build α H through the following procedure: 1. α 0 is the prenex form of α; 2. If α i is an existential formula, then α H is α i ; 3. If α i if of the form ∃x 1 ∃x 2 . . . ∃x n ∀yγ, we introduce a function symbol f , such that: If α i is of the form ∀yγ, we introduce a constant symbol c, such that: We can represent Herbrand's normal form of a prenex formula . Theorem 9 (Herbrand) Let T be a theory without non logical axioms in the language L. Then, for any prenex formula α in L, it holds that: Proof. (Detailed strategy) The procedure for proving Herbrand's theorem follows the steps: 1. If α is an existential formula, the theorem is a corollary of Hilbert-Ackermann's theorem: We suppose that α is of the form ∃x 1 ∃x 2 . . . ∃x n β, with β as an open formula. In this case, ¬α is logically equivalent to ∀x 1 ∀x 2 . . . ∀x n ¬β. Thus, T ⊢ ¬α ↔ ¬β.
3. Let L H be the language L extended with functions used to built α H and T H the theory without logical axioms in the language L H , we should prove that:

4.
Since α H is an existential formula, we obtain a quasi-tautology for the extended language L H . Hence, For this reason, the proof of item 3 finish the proof.
5. To prove item 3, we only show the strategy for the converse implication, that is, T ⊢ α ⇐ T ′ ⊢ α H , since the direct proof is relatively simple.
6. Let T c be the Henkin extension of T , defined in [5, p. 46], and T c+eq be the addition of equivalence axioms 1 of Herbrand [5, p. 52] in T c , we show that: These two fact will be important to the following steps.
7. We suppose that there is a quasi-tautology β 1 ∨ β 2 . . . ∨ β q in L H , with β i being a instance of the matrix α H . We now do a procedure of replacement of functions introduced for α H by constants in L c . More specifically, if α H is of the form and β i if of the form we define the sequence of special constants d(u 1 ), d(u 1 , u 2 ), . . . , d(u 1 , . . . , u k ) for eliminating the functions in the following manner: (a) z i→ is a sequence of terms of the sequence z from the index i onward; (b) z i→j is a sequence of terms of z from the index i up to the index j; (c) (z) i is the i'th term in the sequence z; The constant (d(u 1 )) i is a special constant for the formula begin * a abbreviation that indicates the remaining sequence And, generally, (d(u 1 , u 2 , . . . , u i )) j is a special constant for the formula By successively applying the substitution axiom and modus ponens, we have that, se β ′ i is the formula then ⊢ Tc+eq β ′ i → α. 8. Note that variables can occur in u. However, to ensure the use of the equivalence axiom properties in T c+eq , we need all variables to be replaced by special constants. This will be necessary for the completion of the proof.
So we make a second transformation in the quasi-tautology. Here equivalence axioms play an important role: they ensure that any two equivalent formulas refer to a single constant 2 . With the addition of only the special axioms, we do not have this guarantee, since we could have two distinct elements satisfying the same formula of the form ∃xα.
We introduce distinct special constants for each variable in u, obtaining u ′ . Subsequently, we prove, using the special equivalence axioms, that We usa a similar procedure as in [5, p. 55] to obtain a formula β ′′ 1 ∨ β ′′ 2 . . . ∨ β ′′ q , being β ′′ i the replacement of the functions and variables introduced by the special constants shown above. Recall that the second transformation ensure us that ⊢ Tc+eq β ′′ i → α by the same procedure as in item 7.
We suppose C 1 , C 2 , . . . , C m to be the proof sequence for the quasitautology that uses only identity and equality in T . Thus we prove that the transformation that have led β 1 ∨β 2 . . .∨β k to become β ′′ 1 ∨β ′′ 2 . . .∨β ′′ k preserves tautologically the sequence. Therefore, β ′′ 1 ∨ β ′′ 2 . . . ∨ β ′′ k is tautological consequence of C ′′ 1 , C ′′ 2 , . . . , C ′′ m . It remains to prove that each C ′′ i is a theorem of T c+eq , when C i is an axiom of T . This will result in the proof of β ′′ 1 ∨β ′′ 2 . . .∨β ′′ k in T c+eq . Each C i is an axiom of identity, equality, an instance of identity or an instance of equality. We note that, when we get C ′′ i , we have transformed C i into axioms of identity or equality, unless C i is an instance of identity. This last case, though, is easily proved in T c+eq from the propositions shown in the end of last item.

As
11. Since T c+eq is a conservative extension of the logic in L, we lastly obtain that T ⊢ α.

Finitary proof of equiconsistency
In order to understand the finitary proof, it is important to consider procedures 7 and 8 of the previous section. In them, syntactic transformations are performed to provide the formulas β ′′ i , eliminating the functions introduced to obtain the Herbrand's normal formula.
Each introduction of a function to eliminate a universal quantification is performed independently of the other introductions. In this sense, we can restrict the elimination procedure to certain universal quantifications. In fact, we can eliminate one, some or all functions introduced to obtain the Herbrand's normal form.
For the proof of the equiconsistency theorem, we will use the direct part of Herbrand's theorem to obtain the quasi-tautology. Subsequently, we will make the procedure of the converse proof of Herbrand's theorem restricted to the variables limited to sets.
Before performing this procedure, we will eliminate all universal quantifications that are not restricted to sets in the NBG axioms. We will therefore prove that unrestricted universal quantifications are easily eliminated from the axiomatization presented.

Proof. By substitution, we have
{∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y)} ⊢ ∀z(z ∈ x ↔ z ∈ y) → x = y. And, by generalization, {∀z(z ∈ x ↔ z ∈ y) → x = y} ⊢ ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y). We should now prove that Nevertheless, since we know that, by definition, z ∈ V ↔ ∃w(z ∈ w), then z ∉ V is equivalent to ∀w(z ∉ w). From this, we have that ∀z ∉ V (z ∈ x ↔ z ∈ y) is a tautology. We thus obtain: ◻ Note that we can use a similar procedure to limit other quantifications to sets, whenever we have a universal quantifier ∀z for a formula in which z ∈ x occurs. Although this technique does not work in all cases, for the axiomatization of NBG used in this article the procedure is effective. We will not expose here the corresponding proof for each axiom, for they are all very similar. In this sense, we have the following theorem: Theorem 12 Let α be an axiom of NBG, then there is a formula α ′ such that α ′ is the elimination or restriction (to V ) of all universal quantifications occurring in α and ⊢ α ′ ↔ α.
We modify NBG yet one more time. The unrestricted existential quantifiers can be eliminated from instances of the scheme axiom for classes. We add, for each instance ∀v ∈ V ∃z∀x(x ∈ z ↔ (x ∈ V ∧ α(x, v))), the constant c α and replace the axiom for ∀v ∈ V ∀x ∈ V (x ∈ c α ↔ (x ∈ V ∧ α(x, v))). Subsequently, we replace all other axiom of NBG by the version obtained by the successive application of theorem 12. We call the resulting theory U .
We should prove the following theorem: Theorem 13 For every formula γ, if N BG ⊢ γ, then U ⊢ γ.
Proof. In order to proof this theorem, we should just treat the case in which γ is the scheme axiom for classes. From this and from theorem 12, we obtain easily that U proves all other axioms of NBG.
We take a formula α with n free variables and in which all quantifications are bouded to V . We then prove that (We call this formula θ). By replacement and generalization, we can eliminate the universal quantifications from the scheme axiom for classes. Thus we have Since x and z do not occur in the left side of the implication, then Let θ ′ be the formula ∀v ∈ V ∀x ∈ V (x ∈ c α ↔ (x ∈ V ∧ α(x, v))).
We now eliminate the quantifiers of θ ′ , obtaining Thus, by replacement and modus ponens, Therefore, U ⊢ θ. ◻ We continue by proving the lemma: Lemma 14 Let α be a sentence without variables for classes. If α is a theorem of U , then there is a proof of α in U that is free of unrestricted quantifications.

Proof.
By hypothesis, we have a proof of α in U . Let γ [x] be the conjunction of axioms used in the given proof, in which x is the sequence of variables for classes that occur in the axioms. In this case, be the reduction theorem [5, p. 42]: From Herbrand's theorem provability equivalence, if, and only if, in which T H is the theory obtained by the addition of function symbols in T , according to Herbrand's normal form ∃xθ [x].
By Herbrand's theorem, there is a quasi-tautology β ′ 1 ∨ ... ∨ β ′ m , in which each β ′ i is an instance of β ′ in the language of T H . Let's build the appropriate proof of α in U c+eq , obtained from U by the addition of special constants, special axioms and special axioms of equivalence.
From start, we replace each free variaable in β ′ 1 ∨ ... ∨ β ′ m for special constants. The result is a disjunction of m sentences that are quasi-tautological in U c+eq . This quasi-tautology is the starting point of the proof of α in U c+eq .
Subsequently, we replace the occurences of f 1 (a, b 1 ), . . . , f k (a, b 1 , ..., b k ) for sequences of appropriate special constants. For this, we follow the items 7 and 8 in the proof of Herbrand's theorem.
Let θ ic be the formula ∃z 1 ∀y 1 ...∃z k ∀y k β[t i , z 1 , y 1 , ..., z k , y k ], obtained from β ic by the restating the quantifications ∃z 1 ∀y 1 ...∃z k ∀y k . The closed terms in t i are in U c+eq . Each disjunction θ ic implies the corresponding β ′ ic in U c+eq , as we have seen in the first paragraph of item 9 in Herbrand's theorem proof. To show this, we use only simple properties of implication, of the quantifiers ∃z 1 ...∃z k and the special axioms used in the replacement of the occurrences f 1 (a, b 1 ),...,f k (a, b 1 , ..., b k ) described above. From what we have exposed, it follows that U c+eq ⊢ θ 1c ∨ ... ∨ θ mc , without using quantifications over classes.
using only variations of quantifications that occur in γ[x] → α. Therefore, no quantification for class variables. On the other hand, by the rule of substitution, As θ[t i ] θ ic , we conclude, using tautological consequence, that without using quantification for class variables.
On the other hand, γ[x] is the conjunction of axioms in U c+eq . Each γ[t 1 ] can be proved in U c+eq using only tautological consequences and instances of the substitution rule. Therefore, U c+eq ⊢ α, without using quantification for class variables.
Finally, we observe that any proof of α of U in U c+eq can be transformed in a proof in U of the same α, and that this transformation introduces only quantifications directly related to the special axioms used [5, p. 52]. Since we haven't used special axioms for class variables, the result follows.
◻ From this result, we prove by finitary means the equiconsistency result: Theorem 15 Let α be a sentence with all its quantifiers bounded to sets and such that N BG ⊢ α. Then, ZF ⊢ α.
Proof. Let α be a sentence with all its quantifiers bounded to sets and such that N BG ⊢ α. Thus, by theorem 14, there is a proof in U in which no unbounded quantifications occur. We call this proof sequence Seq.
We make transformations in this proof sequence that preserve tautological consequences, do not affect α and such that the transformed axioms are theorems of ZF. Let x 1 , x 2 , . . . , x k be the free variables that occur in Seq. We add the following initial segment to the proof sequence x 1 ∈ V, x 2 ∈ V, . . . , x k ∈ V , obtaining Seq * 1 .
x ∈ c θ , are replaced by θ(x) The successive application of this transformation to Seq * 1 eliminate all occurrences of c θ , forming the sequence Seq * 2 .
Recall that, for each logical axiom Axiom j in Seq * 1 , we should verify whether Axiom * ZF j is also a logical axiom or a consequence of ZF together with formulas x 1 ∈ V, x 2 ∈ V, . . . , x k ∈ V . In the second case, we replace each axiom Axiom * ZF j in Seq * 2 by the proof sequence of Axiom * ZF j . Hence, we obtain Seq * 3 .
Therefore, all transformations described above do not affect the proof tautologically. However, some instances of the logical axioms may not be logical axioms after the transformation. We now investigate what happen with logical axioms in which constants of U occur.

Substitution axiom
There is no instance of the substitution θ x (c) → ∃xθ in Seq * 1 since unrestricted quantifications do not occur in the initial proof in U .

Equality axiom
(a) If the axiom is of the form then, after the transformation: We show that this formula is theorem of ZF and of the formulas If we have that ∀z ∈ V (z ∈ x 1 ↔ α(z)) ∧ x 2 = y 2 and we suppose that x 1 ∈ x 2 , the, by corollary 2 of the identity theorem in [5, p. 36], we obtain Then, by the substitution axiom in e x 1 ∈ V , Using the variant theorem [5, p. 35], On the other hand, if we suppose that ∃w ∈ V (∀z ∈ V (z ∈ w ↔ α(z)) ∧ w ∈ y 2 ), then, since ∀z ∈ V (z ∈ x 1 ↔ α(z)) and by extensionality in ZF, we prove that ∃w ∈ V (w = x 1 ∧ w ∈ y 2 ). Hence, we obtain that x 1 ∈ y 2 . As, from hypothesis, x 2 = y 2 , we conclude that x 1 ∈ x 2 , finalizing the proof.
(b) x 1 = y 1 ∧ c α = y 2 → x 1 ∈ c α ↔ y 1 ∈ y 2 . The strategy for item b is similar to item a. (c) For this case, we obtain from the transformation: . As x 1 and y 1 are free variables in the formula, the transformed formula is tautological consequence of ∀x 1 ∈ V ∀x 2 ∈ V and instances of identity axioms.
We prove items d, e and f by a simple combination of the strategies used in a and c.
Let's now evaluate what occur to the aximos in U after the transformation * ZF : The instances of extensionality, scheme for classes, replacement for classes and foundations are the only ones we affect by applying * ZF (only in those axioms constants of U occur).

Foundation.
(c θ ≠ ∅ → ∃y ∈ V (y ∈ c θ → c θ ∩ y = ∅)) becomes 4 There is no interpretation of NBG in ZF Before we prove there is no interpretation os NBG in ZF, we show some definitions as propositions: Definition 18 Let V be a model of ZF and M a class V -definable, we say that the model M = (M, ∈ V ) is a V -natural model.

Definition 19
Let M be a model in the language L ZF (the only non logical symbol is membership) and an interpretation I = ⟨U, φ⟩ of L ZF in L ZF (we write ∈ I for φ(∈)), then we define the model M I = (A, ∈ I M ) em L as From this definition, we can easily prove by induction that: Proposition 20 Let M be a model in L ZF and I = ⟨U, ∈ I ⟩ be an interpretation of L ZF in L ZF , then, for all sentences α The following proposition is a strengthening of a result in [?]. They show that if the existence of a transitive model of ZF is consistent with ZF, then there cannot be a set-interpretation of ZF in ZF. Here, we replace that consistency condition for "ZF does not prove the inconsistent sentence for ZF itself".
Proposition 21 Let I = ⟨U, ∈ I ⟩ be an interpretation of ZF in ZF, if ZF ⊢ {x U (x)} is a set, then ZF ⊢ ¬Con(ZF ).
Proof. Take I as in the proposition and suppose ZF ⊢ {x U (x)} is a set.
Let ⌜α⌝ be the Gödel number of the formula α represented in ZF. Since M I is a set for every M ⊧ ZF , we define recursively the set T for each M ⊧ ZF : Notation 22 a([k] = b) is the replacement of the k'th element of the sequence a for b.
⟨⌜α⌝, a⟩ ∈ T if, and only if, a ∈ M I and (1) if α if atomic of the form x i ∈ x j , ⟨a i , a j ⟩ ∈ (∈ M I ) (2) if α is of the form β ∧ γ ∶ ⟨⌜β⌝, a⟩ ∈ T and ⟨⌜γ⌝, a⟩ ∈ T (3) if α is of the form ¬β ∶ ⟨⌜β⌝, a⟩ ∉ T (4) if α is of the form ¬∃x k β ∶ ⟨⌜β⌝, a([k] = b)⟩ ∈ T for some b ∈ M I By finite induction over the formula complexity, we prove that M ⊧ ⟨⌜ϕ⌝, a⟩ ∈ T if, and only if, M I ⊧ ϕ(a) Take P r ZF (x, y) to be the provability predicate for ZF defined in ZF and representing the statement "x is the number of the proof y". Then, we say that T h(ZF ) = {y ∃xP r ZF (x, y)}. Hence, since M I ⊧ ZF , M ⊧ T h(ZF ) ⊆ T . As M I ⊭ ∅ ∈ ∅, we have ⌜∅ ∈ ∅⌝ ∉ T . From this, we obtain M ⊧ ⌜∅ ∈ ∅⌝ ∉ T h(ZF ). Once M is arbitrary, by the completeness theorem, if ZF has a model, then ZF ⊢ ⌜∅ ∈ ∅⌝ ∉ T h(ZF ). This is an absurd by Gödel's incompleteness theorem. Thus, ZF ⊢ ¬Con(ZF ). We show that the desired result is a consequence of the reflection theorem and the fact that NBG is finitely axiomatizable: Theorem 25 There is no interpretation of NBG in ZF.

Proof.
Suppose there is an interpretation I of NBG in ZF. Since the number of axioms in NBG can be said to be finite, there is a formula α that is equivalent to the conjunction of all NBG's axioms: α is Axiom 1 ∧ Axiom 2 ∧ . . . ∧ Axiom n .
Thus, NBG ⊢ α and, from the interpretation theorem for first-order logic, ZF ⊢ α I .
Suppose V ⊧ ZF , then V ⊧ α I . By the reflection theorem, there is an ordinal a such that V a ⊧ α I . It follows that V I a ⊧ α. Since V a is a set, we obtain that the domain in V I a is also a set. We define the model V * in L ZF : 1. The domain D in V * is such that D = {x V I a ⊧ ∃y(x ∈ y)}.
Since NBG proves the restriction to sets of all ZF axioms, we have that V * ⊧ ZF . Since the domain V I a is a set, it follows that D is also a set. It means that ZF ⊢ Con(ZF ), absurd by the incompleteness theorem. ◻