In a sound proof system, every provably total function is indeed total, but the converse is not true : in every first-order proof system that is strong enough and sound ( including Peano arithmetic ), one can prove ( in another proof system ) the existence of total functions that cannot be proven total in the proof system.
32.
During the second year of operation, students purchased building materials from a local Liberty merchant ( with funds probably provided from Dr . Bowman's personal account ) and installed a sound proof, windowed partition between the broadcast booth and the transmitter, finally eliminating transmitter fan noises that had been so noticeable when a microphone was open in the booth.
33.
In a sound proof system, every provably total Turing machine is indeed total, but the converse is not true : informally, for every first-order proof system that is strong enough ( including Peano arithmetic ), there are Turing machines which are assumed to be total, but cannot be proven as such, unless the system is inconsistent ( in which case one can prove anything ).
