The following theory is a class theory, where all classes are either classes of ordinals, or relations between classes of ordinals, that is, classes of ordered pairs of Kuratowski. However, the size of his universe is poorly inaccessible. Ordinals are defined as von Neumann ordinals. The theory is formalized in a logic of first order with equality and belonging.

**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, so all closures of: $$ exists x forall y (y in x leftrightarrow exists z (y in z) land phi) $$; are axioms.

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

*To define:* $ langle alpha beta rangle = { { alpha }, { alpha, beta } $

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

**Reports:** $ forall text {ordinals} alpha beta exists x ( langle alpha, beta row in x) $

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

**Cut:** $ ORD text {is weakly inaccessible} $

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

/ Definition of the completed theory.

Now, this theory can clearly define various arithmetic operations extended on elemental ordinals. This also proves transfinite induction on elemental elements. In a sense, this can be considered as an arithmetic extensible to the infinite world. Of course $ PA $ is interpretable in the finite segment of this theory.

In that response, Nik Weaver in his response raised the concern of ZFC as being arithmetically unfounded.

My question is this: assuming this theory to be consistent, is the problem of its arithmetic calculation identical to that of ZFC?

The reason for this question is that it seems to me that the above theory is only a naive extension of numbers to the infinite world, it has no power axiom nor equivalent. We can say that this theory is in a way purely mathematical in the sense that it only concerns numbers and their relations. Would this raise the same kind of suspicion about the non-arithmetic calculated with ZFC.

My reasoning on this point is that, in general, when one raises the concern about the arithmetical inadequacy of a theory, especially if this theory is welcomed by mathematicians working in set theory and the foundations, there must be a technical or intuitive argument behind this suspicion, otherwise this suspicion would be unfounded. Suspicion should not depend solely on the strength of the theory in question. Otherwise, we would define no stronger theory than $ PA $ based on such concerns.

According to Nik Weaver's answer, it seems to me that his concern is based on the fact that ZFC does not intuitively capture a clear concept. However, this theory is based on an intuitive concept generally similar to that which defines the arithmetic of finite sets. He extends it intuitively very clearly, the higher ordinals are *defined* precedents successively, and it does not generally seem to be so different from the intuitive foundations of arithmetic in the finite world. The question is therefore whether this theory is still the prey of the arguments on which the concerns about the arithmetical non-compliance of ZFC are based.