[FOM] Lucas, Penrose, and the Church-Kleene ordinal

[Torkel Franzen]

Timothy Chow says:

 >Here's where w_1^CK comes in.  Unless I'm confused (and if I am, please
 >enlighten me), we get "minimal" examples of S by starting with Q and
 >iteratively adjoining stronger and stronger consistency statements,
 >until we're forced to stop when we finally get a *non-recursive* CFA
 >and therefore can't apply Goedel's theorem any more. 

  No, we're never "forced to stop" in a theoretical sense. Any
effectively axiomatizable theory that we recognize as an iterated
consistency extension of Q we can also extend by adding one more
consistency statement.