Thanks for the feedback John; which seems to be consistent with what I (think I) know.
(PS. Hopefully I will see you tomorrow at the NYCJUG meeting) > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:programming- > [EMAIL PROTECTED] On Behalf Of John Randall > Sent: Monday, July 09, 2007 5:02 PM > To: Programming forum > Subject: RE: [Jprogramming] Learning Mathematics versus Learning J == > learn by theory versus learn by practice? > > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
