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>
To say that is about as stupid as to say something like "Do you believe in a screwdriver." It is a tool that you use, just like set theory. Mathematics is saying A=>B, there is nothing that forces anyone to "believe" A is true. If people just say "B", then they are implicitly assuming that everyone knows there is some axiom system in the background and most people would assume something in terms of ZFC or some such and that B is really just a result "ZFC => B." However, to state that for just about every result in the literature would become cumbersome.
Most mathematicians would, in my opinion, "agree" that ZFC are most probably consistent, and most probably the best basis (currently) for modeling the real world. Pragmatic acceptance is all that is required to work with ZFC. Belief is not neccessary (but of course not disallowed). And of course most mathematicians don't really use ZFC, but rather naive set theory. All that we need ZFC for is to show that most naive set reasoning probably does follow from a few consistent axioms.