>>13899841Set theory is NOT just a language in which to express mathematical results. It's an extremely strong set of assumptions about the way mathematical world, including natural numbers, work. By accepting different kinds of set theory, you can change the truth status of many first order arithmetic propositions, even many first-order complexity
algorithm halting statements. Much of the assumptions of set theory are as of yet completely unjustified.
>>13899918The axiom of foundation is a very funny case. Mathematicians claim to understand the concept of the set through the cumulative hierarchy. The cumulative hierarchy is only exhaustive because it is claimed to be by the axiom of foundation. Typically when one defines some sort of mathematical structure, one then investigates and proves its properties. In the case of the structure of the set theoretic universe it's the other way around: mathematicians list a bunch of properties that they think this structure should satisfy as axioms, and then they use these axioms to claim to understand the structure. Such is the backwards thinking when you fall for set theory.
>>13899952>>13899995Complex numbers were controversial for the same reason that set theory was: neither was rigorously defined, people just listed a bunch of properties they want them to satisfy (in the case of complex numbers: square equals -1, in the case of set theory, the primitive form of ZFC). The difference is, that by now complex numbers have been rigorously defined in numerous ways, but sets are still undefined.
>>13899985Not all axioms are created equal, even logically consistent ones. Even a dirty formalist like you must accept that PA + Con(PA) and PA + not Con(PA) are not equally sound. One is obviously flawed for mathematical practice, even though both are perfectly consistent.
