In Gödel's collected works vol. 2 one has the detailed argument by transfinite induction up to the first inaccesible ordinal. Interesting to have this as guidemap.
By "first inaccessible ordinal" you mean epsilon_0? (Probably you mean it in the sense of the first one inaccessible by addition, multiplication and exponentiation.) Because there is also a "first recursively inaccessible ordinal", which is much larger (also, not recursive: it's used to construct, by collapsing, the ordinal of Delta12-CA+BI, which is also much larger though recursive).