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definability by formulas in first-order logic, e.g. as explained at https://en.wikipedia.org/wiki/Definable_set, or as in J. Robinson's first-order definition of the integers in the field of rationals

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What ordinals are definable relations in Peano Arithmetic?

The computable ordinals---that is, the ordinals below $\omega_1^{CK}$---are, by definition, represented by computable relations, all of which can be represented by formulas in PA, and indeed, even by …
Henry Towsner's user avatar