Geodesic
Arithmetical transfinite induction and hierarchies of functions
Fundamenta Mathematicae, Tome 141 (1992) no. 1, pp. 1-20

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DOI

We generalize to the case of arithmetical transfinite induction the following three theorems for PA: the Wainer Theorem, the Paris-Harrington Theorem, and a version of the Solovay-Ketonen Theorem. We give uniform proofs using combinatorial constructions. 
Z. Ratajczyk. Arithmetical transfinite induction and hierarchies of functions. Fundamenta Mathematicae, Tome 141 (1992) no. 1, pp. 1-20. doi: 10.4064/fm-141-1-1-20
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