User blog comment:B1mb0w/The S Function Version 2/@comment-5529393-20160710173738/@comment-5529393-20160712121200

Hmm, I'm still not sure that the definition of "arbitrary S function" works. It looks like a_0, b_0 and c_0 can be arbitrary S functions, but they are being used before arbitrary S functions have been defined. So the definition looks circular.

Also, I'm not following your argument that we can use the relations < and = before the S function has been defined. It looks to me that the definitions of < and = depend on the full set of S functions, so defining S functions based on < and = also looks circular.