# Can set theory be interpreted in infinite arithmetic?

Is the following system of infinite arithmetic consistent?

If so, can it interpret $$\sf ZFC$$?

Language: first order logic

Primitives: $$\operatorname{Card}, <, + , \times,\text{^}$$

where $$\operatorname{Card}$$ is one place predicate symbol denoting "is a cardinal".

We'll denote this language by the language of arithmetic.

Areflexive: $$x \not < x$$

Transitive: $$x < y < z \to x < z$$

Connected: $$x \neq y \leftrightarrow [x < y \lor y < x]$$

Well-Founded: if $$\phi$$ is a formula, then: $$\phi(x) \to \exists a: \phi(a) \land \forall b: \phi(b) \to b \not < a$$

Cardinality: if $$\phi$$ is a formula in two free variables; then: $$\operatorname{Card}(x) \land y < x \land [\phi: prior(y) \to prior(x), \phi \text { is one-one}] \\ \to \phi \text { is not surjective }$$

Successor Cardinals: $$\forall x \, \exists y: \operatorname{Card}(y) \land y > x$$

Replacement: $$[\phi: \psi \to prior(l), \phi \text{ is one-one}] \to \exists k: \forall x (\psi(x) \to k > x)$$

Define: $$x=0 \iff x=\min_{(<)} y: y=y$$

Define: $$S(x)=y \iff y= \min_{(<)} z: z > x$$

Infinity: $$\exists l \neq 0: \forall r < l \exists s: r < s < l$$

Addition: $$a + 0 = a \\ a + S(b) = S(a+b) \\ a + b = b + a \\ \lambda + \zeta = \lim (\lambda + \alpha)_{\alpha<\zeta} \\\text{; for limits } \lambda, \zeta \text{; and } \lambda \geq \zeta$$

Multiplication: $$a \times 0 = 0 \\ a \times S(b) = a + (a \times b) \\ a \times b = b \times a \\ \lambda \times \zeta = \lim (\lambda \times \alpha)_{\alpha < \zeta} \\\text{; for limits } \lambda, \zeta \text{; and } \lambda \geq \zeta$$

Exponentiation: $$a \text{^} 0 = 1 \\ a \text{^} S(b) = a \times (a \text{^} b) \\ a \text{^} \lambda= \lim (a^\kappa)_{\kappa < \lambda} \text{; for limit }\lambda$$

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Terminology:

• $$[\phi: \psi \to \pi, \phi \text { is one-one }]$$ is the following formula: $$\forall x \,[ \psi(x) \to \exists y: \pi(y) \land \phi(x,y)] \land \\ \forall a,b,c,d \, (\phi(a,b) \land \phi(c,d) \to [a=c \leftrightarrow b=d] )$$

• $$prior(l)$$ is the formula $$(x < l)$$

• $$( \phi \text { is not surjective})$$ in the above expression, it is exactly the formula: $$\exists b < x: \forall a \, [a < y \to \neg \phi(a,b)]$$

• At a glance it's definitely consistent relative to $\mathsf{ZFC}$, since (unless I'm missing something) it's straightforwardly interpretable in it. Commented Sep 25, 2023 at 17:32
• @NoahSchweber, I was thinking of interpreting $\sf ZFC$ in this system through defining ordered pairs, which I think its feasible, then define well founded extensional graphs, but the problem is in defining the latter, I need a kind of collective function that is definable in arithmetic. Commented Sep 25, 2023 at 18:01
• Takeuti's "A Formalization of the Theory of Ordinal Numbers" might be relevant to your question. He proved that his theory of ordinals interprets $\mathsf{ZFC}$, although his theory is different from yours. Commented Sep 25, 2023 at 22:03