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I had pointed out earlier that the question as asked has a negative answer, in light of the counterexample provided by my answer to your previous question.

I had pointed out earlier that the question as asked has a negative answer, in light of the counterexample provided by my answer to your previous question.

So let me provide a different counterexample here in the infinite case, using natural arithmetic. The principle (**) remain inconsistent with ZFC, even if one should use natural arithmetic.

Claim. $T:\text{Ord}\to\mathbb{N}$ is a homomorphism with respect to natural addition and multiplication. That is, $$T(\alpha+\beta)=T(\alpha)+T(\beta)$$ $$T(\alpha\cdot\beta)=T(\alpha)\cdot T(\beta),$$ where we use natural arithmetic on the ordinals.

So let me provide a different counterexample here in the infinite case, using natural arithmetic. The principle (**) remains inconsistent with ZFC, even if one should use natural arithmetic.

Claim. $T:\text{Ord}\to\mathbb{N}$ is a homomorphism with respect to natural addition and multiplication. That is, $$T(\alpha+\beta)=T(\alpha)+T(\beta)$$ $$T(\alpha\cdot\beta)=T(\alpha)\cdot T(\beta),$$ where on the left-hand side, we use natural arithmetic on the ordinals, and on the right hand side is the usual arithmetic on $\mathbb{N}$.

So let me provide a different counterexample here in the case of natural arithmetic.

So let me provide a different counterexample here in the infinite case, using natural arithmetic. The principle (**) remain inconsistent with ZFC, even if one should use natural arithmetic.