The following theory is a class theory, where all classes are either classes of ordinals, or relations between classes of ordinals, i.e. classes of Kuratowski ordered pairs of ordinals, or otherwise classes of unordered pairs of ordinals. However, the size of its universe is weakly inaccessible. Ordinals are defined as von Neumann ordinals. The theory is formalized in first order logic with equality and membership.

**Extensionality:** $ \forall z (z \in x \leftrightarrow z \in y) \to x=y$

**Comprehension:** if $ \phi$ is a formula in which the symbol $ “x”$ is not free, then all closures of: $ $ \exists x \forall y (y \in x \leftrightarrow \exists z(y \in z) \land \phi)$ $ ; are axioms.

**Ordinal pairing:** $ \forall \text{ ordinals } \alpha \beta \ \exists x (\{\alpha,\beta\} \in x) $

*Define:* $ \langle \alpha \beta \rangle = \{\{\alpha\},\{\alpha,\beta\}\}$

**Ordinal adjunction:**: $ \forall \text { ordinal } \alpha \ \exists x (\alpha \cup \{\alpha\} \in x)$

**Relations:** $ \forall \text{ ordinals } \alpha \beta \ \exists x (\langle \alpha, \beta \rangle \in x)$

**Elements:** $ \exists y (x \in y) \to ordinal(x) \lor \exists \text{ ordinals } \alpha \beta \ (x=\langle \alpha,\beta \rangle \lor x=\{\alpha,\beta\})$

**Size:** $ ORD \text { is weakly inaccessible}$

Where $ ORD$ is the class of all element ordinals.

/Theory definition finished.

Now this theory clearly can define various extended arithmetical operations on element ordinals. Also it proves transfinite induction over element ordinals. In some sense it can be regarded as stretching arithmetic to the infinite world. Of course $ PA$ is interpretable in the finite segment of this theory.

In this posting Nik Weaver in his answer raised the concern of ZFC being arithmetically unsound.

My question: assuming this theory to be consistent, is the concern of it being arithmetically unsound is the same as that with ZFC?

The motive for this question is that it appears to me that the above theory is just a naive extension of numbers to the infinite world, it has no power set axiom nor the alike. One can say that this theory is in some sense purely mathematical in the sense that it’s only about numbers and their relations. Would this raise the same kind of suspicion about arithmetic unsoundness that is raised with ZFC.

My reasoning about that is that generally speaking when one raises the concern of arithmetic unsoundness of some theory, especially if that theory is well received by mathematicians working in set theory and foundations, then there must be some technical or intuitive argument behind that suspicion, otherwise that suspicion would be unfounded. The suspicion must not depend merely on the strength of the theory in question. Otherwise we’d not define any theory stronger than $ PA$ based on such concerns.

From Nik Weaver’s answer it appears to me that his concern is based on ZFC not capturing a clear concept intuitively speaking. Now this theory is based on an intuitive concept that is generally similar to the one behind defining arithmetic for finite sets. It extends it in a very clear intuitive manner, higher ordinals are *defined* from prior ones in successive manner, and it doesn’t generally feel to be so different from the intuitive underpinnings of arithmetic in the finite world. So the question here is about if this theory still fall a prey to the arguments upon which the concerns about arithmetic unsoundness of ZFC are based.