TITLE: Difference between undecidable statements in set-theory and number theory?
QUESTION [11 upvotes]: Do all statements about the integers have a definite truth value?
For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to disprove it. So the independence proof is also a proof. Godel's theorem only claims there are true statements that are unprovable.
What is it that makes statements about the integers have a clear true/false-value (by some higher-order reasoning), but statements about sets such as AC and CH dont?
Does the higher-order reasoning just become more and more diffuse (or subjective) or is there some fundamental leap between undecidable statements about the integers vs sets?

REPLY [6 votes]: What is it that makes statements about the integers have a clear true/false-value (by some higher-order reasoning), but statements about sets such as AC and CH dont?

The key difference is that

The set of natural numbers is defined by a sort of "minimailty principle": the natural numbers are, up to isomorphism, the smallest set containing 0 and closed under the function $f(x) = x+1$. 
The collection of all sets is defined by a sort of "maximality principle": it consists of all sets. 

This difference makes the natural numbers concrete in a certain way the the collection of all sets is not. 
If we try to take the corresponding "minimality" principle for sets, we end up with the constructible universe $L$, or with some sort of minimal submodel of it, for which we can answer many of the questions like AC and CH that are independent of ZFC.

REPLY [3 votes]: What is it that makes statements about the integers have a clear true/false-value (by some higher-order reasoning), but statements about sets such as AC and CH don't?

This is about the logical complexity of the formulas. We can have statements about integers that uses very high logical complexity (e.g. by quantifying over all subsets of natural numbers). But I guess that is not what you mean by a statement about integers, you probably mean a statement is in arithmetical hierarchy. Similarly, a statement about sets can be very low logical complexity, e.g. if the set is definable by a low complexity statement. What is important here is not the free variables of the statement (the objects the statement is about) but the logical complexity of the statement.
Why arithmetical statements seem more concrete and well-defined (at least to some) than say analytical statements?
One reason is that arithmetical statements do not quantify over infinite objects (which are not as intuitive as finite objects). Another objection is that the analytical statements are impredicative because a definition of the object quantifies over the set of objects that the object itself belongs to (similar to Russell's paradox). There are various views in philosophy of mathematics about which statements are really well-defined and have a definite truth value and which statements do not.

Does the higher-order reasoning just become more and more diffuse (or subjective) or is there some fundamental leap between undecidable statements about the integers vs sets?

It is not about the order of reasoning. The difference is that infinite objects are not like finite objects (objects that at least in principle can be represented physically). I can show you 2 apples but I cannot show you $\aleph_0$ apples. Our intuition about infinite objects is not as good as our intuition about finite ones, e.g. if we try to do the set theory in Cantor's naive way we will ran into paradoxes. Two distinct views about infinite objects are:

top-down: every possible object is OK (unless something goes wrong), like Cantor's naive set theory, or Frege's system, which have a Comprehension Axiom (which gives objects out of statements).
bottom-up: only objects we can build by some methods are OK, like current axiomatic set theories like ZFC, which do not have a Comprehension Axiom.

If we are in the first framework, then we need to decide what kind of properties are acceptable definitions of infinite objects. Is Analytical Comprehension OK? What about even more complex Comprehension Axioms?
If we are in the second framework, we need to discuss which methods of building are OK. E.g. is (classical) Power Set axiom OK? What about axiom about existence of an Infinite Set? (Note that without Infinite Set axiom, set theory is similar to PA). 
And of course, there are other philosophical views where one rejects that infinite sets and objects exist (some completely reject them and some believe that they do not exist as completed objects, i.e. we only have potentially infinite objects which are limits of specific sequence of finite objects).
Philosophy of mathematics is a big area, if you are interested there nice introductionary articles on Wikipedia and also on SEP, you can find also nice books, surveys, and papers on the topic by checking the reference part of those articles.