Jose Mario Quintana wrote: > Yet, as far as I understand, a consequence of Godel's Incompleteness > Theorem > (found by himself) states that the consistency of the ordinary arithmetic > cannot be proven (at least, an effective proof cannot exist). In other > words, believing in the consistency of the arithmetic is, ultimately, an > act > of faith. (Then again, what is not?) > > To complicate matters, at least for me, there is a result by Tarski > implying > that (an axiomatic version of) the field of real numbers is complete. I > am > not familiar with its proof and I do not know how to reconcile it with the > former statement. Can you, or anybody else, provide some enlightenment? > (My > only guess is that the above mentioned axiomatic description of the field > of > real numbers is insufficient to imply the axioms of the ordinary > arithmetic > of the natural numbers presumed to be embedded.) >
I was somewhat unclear about what I said, so here's an explanation of what I think I know. A system is consistent if it cannot derive a contradiction. A system is complete if every statement can be proved true or false. First order logic is complete. Second order logic does not admit a complete proof theory. A system containing arithmetic on the natural numbers cannot demonstrate its own consistency. However, this may be done using a larger model. For example, arithmetic can be proved consistent using set theory, if set theory is consistent. But then you cannot prove set theory is consistent using only set theory... Tarski's axiomitization of the reals is a second-order theory, and is complete. However, while the natural numbers are a subset, the theory is too weak to identify them. Even if it could, the proof would have to descend to the natural numbers. Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
