Forcing for Hat Inductive Definitions in Arithmetic – One of the Simplest Applications of Forcing –. Sato, K. Mathematical Logic Quarterly, 60(4-5):314–318, 2014.
Forcing for Hat Inductive Definitions in Arithmetic – One of the Simplest Applications of Forcing – [pdf]Paper  doi  abstract   bibtex   
By forcing, we give a direct interpretation of $\widehat{\mathsf{ID}}_\omega$ into Avigad's $\mathsf{FP}$. To the best of the author's knowledge, this is one of the simplest applications of forcing to ``real problems''.

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