by Dennis J. Darland
February, 2021
I have always wondered how it could be that adding the Goedel statement to PM’s axioms would not change the functions computable in the resulting PM. If you are not careful you might think the resulting system inconsistent – Goedel’s statement is now provable. But Goedel’s proof would have to be adjusted for the new axiom. There would now be a new Goedel statement – different. But how could all the same functions be non-computable in the two systems? I had always assumed this (or an equivalent to it) had been checked – the book said Church’s thesis was proven for all known systems & it was a thesis because not all possible systems could be checked. But then the book said Church’s thesis was used to prove the halting problem. I had thought the halting problem was evidence for the thesis. So it all seems circular. The book I am referring to is Computability and Logic, Third Edition, by George S. Boolos and Richard C. Jeffrey. I quoted it at: https://eclectic-philosopher.com/e_pluribus_unum/1903-1995-alonzo-church/