How can the Continuum Hypothesis be independent of the ZFC axioms? Why does the lack of “explicit” examples of sets with a cardinality between that of the naturals and that of the reals not guarantee that there are no examples at all? What would an “implicit” example even mean?
It means that you can’t reach a contradiction by starting with “Let S be a set of intermediate cardinality” and following axioms of ZFC.
All the things you know and love doing with sets —intersection, union, choice, comprehension, Cartesian product, power set — you can do those things with S and nothing will go wrong. S “behaves like a set”, you’ll never catch it doing something unsetlike.
Another way to say this is: There is a model of ZFC that contains a set S of intermediate cardinality. (There is also a model of ZFC that doesn’t. And I’m sympathetic to the view that—since there’s no explicit construction of S -we’ll never encounter an S in the wild and so the model not including S is simpler and better.)
Caveat: All of the above rests on the usual unstated assumption that ZFC is consistent! Because it’s so common to leave it unstated, this assumption is questioned less than maybe it should be, given that ZFC can’t prove its own consistency.
At least one mathematician (I forget his name) considers V=L to be a reasonable axiom to add. Informally put, it says that nothing exists except the things that are required to exist by the axioms. ZF + V=L implies choice, the generalised continuum hypothesis, and many other things. His argument is that just as we consider the natural numbers to be the numbers intended to be generated by the Peano axioms, i.e. the smallest model, so we should consider the constructible universe L to be the sets intended to be generated by the ZF axioms. The axioms amount to an inductive definition, and the least fixed point is the thing they are intended to define. One can think about larger models of ZF, just as one can think about non-standard natural numbers, but L and N are respectively the natural models. I don’t know how popular this view is.
In Peano arithmetic, the induction axiom (not axiom schema) basically says ”… and nothing else is a natural number”. It can only be properly formulated in second-order logic, and the result is that Peano arithmetic becomes “categorical”, which means it has only one (the intended) model up to isomorphism. The real or complex number systems and geometry also have categorical axiomatizations. Standard (first-order) ZFC is not categorical, since it allows both for models that are larger than intended (like first-order Peano arithmetic) and smaller than intended (unlike first-order Peano arithmetic). However, second-order ZFC is also not categorical, although I think it rules out some part of the non-standard models. But your description of the theory ZF+V=L sounds like this theory (i.e. a second-order version) would indeed be categorical. Though presumably this would be somewhat of a big deal but is nowhere mentioned in the Wikipedia article. So I guess the theory probably is still not categorical.
Your “simpler is better” is hard to apply. One way of thinking about models where there are no intermediate cardinals isn’t that S doesn’t exist. But that T, a mapping from S to either the naturals or the reals, does exist.
And T will also be something you can’t explicitly construct.
Also, the axiom of choice basically says “there exists loads of sets that can’t be explicitly constructed”.
It means that you can’t reach a contradiction by starting with “Let S be a set of intermediate cardinality” and following axioms of ZFC.
All the things you know and love doing with sets —intersection, union, choice, comprehension, Cartesian product, power set — you can do those things with S and nothing will go wrong. S “behaves like a set”, you’ll never catch it doing something unsetlike.
Another way to say this is: There is a model of ZFC that contains a set S of intermediate cardinality. (There is also a model of ZFC that doesn’t. And I’m sympathetic to the view that—since there’s no explicit construction of S -we’ll never encounter an S in the wild and so the model not including S is simpler and better.)
Caveat: All of the above rests on the usual unstated assumption that ZFC is consistent! Because it’s so common to leave it unstated, this assumption is questioned less than maybe it should be, given that ZFC can’t prove its own consistency.
At least one mathematician (I forget his name) considers V=L to be a reasonable axiom to add. Informally put, it says that nothing exists except the things that are required to exist by the axioms. ZF + V=L implies choice, the generalised continuum hypothesis, and many other things. His argument is that just as we consider the natural numbers to be the numbers intended to be generated by the Peano axioms, i.e. the smallest model, so we should consider the constructible universe L to be the sets intended to be generated by the ZF axioms. The axioms amount to an inductive definition, and the least fixed point is the thing they are intended to define. One can think about larger models of ZF, just as one can think about non-standard natural numbers, but L and N are respectively the natural models. I don’t know how popular this view is.
In Peano arithmetic, the induction axiom (not axiom schema) basically says ”… and nothing else is a natural number”. It can only be properly formulated in second-order logic, and the result is that Peano arithmetic becomes “categorical”, which means it has only one (the intended) model up to isomorphism. The real or complex number systems and geometry also have categorical axiomatizations. Standard (first-order) ZFC is not categorical, since it allows both for models that are larger than intended (like first-order Peano arithmetic) and smaller than intended (unlike first-order Peano arithmetic). However, second-order ZFC is also not categorical, although I think it rules out some part of the non-standard models. But your description of the theory ZF+V=L sounds like this theory (i.e. a second-order version) would indeed be categorical. Though presumably this would be somewhat of a big deal but is nowhere mentioned in the Wikipedia article. So I guess the theory probably is still not categorical.
Your “simpler is better” is hard to apply. One way of thinking about models where there are no intermediate cardinals isn’t that S doesn’t exist. But that T, a mapping from S to either the naturals or the reals, does exist.
And T will also be something you can’t explicitly construct.
Also, the axiom of choice basically says “there exists loads of sets that can’t be explicitly constructed”.