Torkel Franzen argues that all the axioms of ZFC are manifestly true the logic apparatus is truth preserving therefore all is good and the system is consistent.
First of all if this is true the the anti-machinists such as Lucas or Penrose are right because Franzen has just made an argument a machine cannot do.
So the axioms are manifestly true and the rules are truth preserving. The argument seems impeccable. What could possibly be wrong with it? For example is it possible that the logical apparatus contributes some spurious truths in addition to preserving them? This
(x)((x+3 < x) --> (x = x+4)) (1)
does not look manifestly true to me. Where did it come from? From the axioms? At this point discussion with the indoctrinated people becomes difficult. They just repeat that (1) is true under all interpretations. They are not able to see the problem. In fact this has nothing to do with any interpretations. The same problem occurs at the propositional level:
(P & ~P) --> Q (2)
is notoriously counter-intuitive. It is called PARADOX of material implication, and it motivated research into relevance logics. So don't tell me that it is all based on manifest truth. In fact I have shown in another thread https://groups.google.com/forum/?hl=en&fromgroups#!topic/sci.logic/lDJcgOg4vco that the proof that the truths of first order arithmetic are not recursively enumerable is NOT likely to hold if we use Strawson-like semantics.