cbr...@cbrownsystems.com wrote: > Karl Malbrain wrote: > > Patricia Shanahan wrote: > > > Karl Malbrain wrote: > > > > Patricia Shanahan wrote: > > > >> Ultimately, I don't think the subject of this thread even asks the right > > > >> question. It should be "Set theory: Should you use?". I don't even know > > > >> what it means to believe set theory. > > > >> > > > > > > > > Believing means to agree with the axioms of set theory. > > > > > > > > intransitive verb > > > > 1 a : to have a firm religious faith b : to accept as true, genuine, or > > > > real <ideals we believe in> <believes in ghosts> > > > > > > > > karl m > > > > > > > > > > That pushes it back to the question of what does it mean to "agree with > > > the axioms". > > > > agree: > > 1. To harmonize in opinion, statement, or action; to be in > > unison or concord; to be or become united or consistent; > > to concur; as, all parties agree in the expediency of the > > law. > > > > You agree with the given system of axioms that negate the > > inconsistencies of the previous system. > > > > > It could either mean "agree that they appear to be good, workable, > > > axioms", or "agree that they are true, in some absolute sense that would > > > make contradictory sets of axioms false". > > > > Yes, sets of axioms that are contradictory within themselves make an > > inconsistent system. So both are true. > > > > > I agree with them in the first sense, but not the second. > > > > I don't see how. > > > > Let theory A be geometry using Euclid's rules, including the parallel > postulate; i.e., loosely speaking, that parallel lines don't intersect. > > > Let theory B be Euclid's postulates, excluding the parallel postulate, > and instead including the postulate that every pair of distinct > parallel lines intersect at two points. > > Theory A is useful in certain cases; theory B is useful in other cases. > Why must one be "true" and the other "false", simply because the two > theories contradict each other?
Yes, the two systems within themselves must hold their counterparts false.
> > > > I do not believe in them as a matter of religious or similar belief. For > > > example, I would not be particularly disturbed if I heard tomorrow that > > > someone competent to evaluate proofs had found a proof of inconsistency > > > of ZF. > > > > religion: > > 4. Strictness of fidelity in conforming to any practice, as > > if it were an enjoined rule of conduct. > > > > Licensed plumbers conform to a strict practice with great fidelity, "as > if" it were an enjoined rule of conduct. Do you claim that "Plumbing" > is a religion?
enjoin: To lay upon, as an order or command; to give an injunction to; to direct with authority; to order; to charge
Yes, more so than not. Plumbers don't accept outsiders to enjoin the trade. You describe the problem you're having to a plumber and he fixes it (or not).