Transfinite Recursion Theorem without Axiom of Replacement implies Counting Theorem

Theorem

Let the Transfinite Recursion Theorem (Formulation $5$) be accepted as axiomatic.

Let the Axiom of Replacement not be accepted.

Then the Counting Theorem holds.

That is, it is possible to prove the Counting Theorem by using the Transfinite Recursion Theorem (Formulation $5$) as an axiom in place of the Axiom of Replacement.

Proof

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Exercise $5.1$