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?
> > 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?