It is true because the discrete world has the potential for more fine structure.
As an example showing that it is true, read https://math.stackexchange.com/a/362840/6708 explaining how it is possible that the real numbers are complete (all first order statements about them are either provably true or false) while the integers are famously incomplete (no set of axioms can prove everything about them).
ALL of the philosophically tricky notions that you list are only tricky when you mix in discrete notions like "integers" in your construction. And given that discrete mathematics gives us things like the Halting problem and incompleteness WITHOUT continuity being involved, shows that it is discrete mathematics that is harder.
> As an example showing that it is true, read https://math.stackexchange.com/a/362840/6708 explaining how it is possible that the real numbers are complete (all first order statements about them are either provably true or false) while the integers are famously incomplete (no set of axioms can prove everything about them).
> ALL of the philosophically tricky notions that you list are only tricky when you mix in discrete notions like "integers" in your construction.
That may be so, but people generally don't use RCF exclusively. Real analysis always looks at the real numbers as an extension of the natural numbers, you can't go anywhere without sequences and limits. So it seems disingenuous to say that continuous mathematics is easier just because RCF is complete.
> And given that discrete mathematics gives us things like the Halting problem and incompleteness WITHOUT continuity being involved, shows that it is discrete mathematics that is harder.
While this is true, it's also true that a purely additive theory of the natural numbers is complete and several other theories of discrete structures are too (for example, while group theory itself is not complete, it's certainly completable, unlike Peano Arithmetic).
Also, the halting problem and incompleteness aren't where weirdness ends. There's a whole other range of weirdness that happens only once you add in uncountable infinities, such as Skolem's paradox, the Banach-Tarski paradox or the undecidability of the continuum problem.
As an example showing that it is true, read https://math.stackexchange.com/a/362840/6708 explaining how it is possible that the real numbers are complete (all first order statements about them are either provably true or false) while the integers are famously incomplete (no set of axioms can prove everything about them).
ALL of the philosophically tricky notions that you list are only tricky when you mix in discrete notions like "integers" in your construction. And given that discrete mathematics gives us things like the Halting problem and incompleteness WITHOUT continuity being involved, shows that it is discrete mathematics that is harder.