> The propositions that are undecidable are NOT AT ALL connected to the propositions that are independent.
I'm not quite sure what you mean by undecidable propositions, but I assume it to mean something like the following?
Propositions which must be true or must be false (i.e. are satisfied/not satisfied respectively by all models which satisfy the overall theory), but for which there is no finite proof that they hold.
In contexts where Godel's Completeness theorem (not incompleteness theorem) holds undecidable propositions are exactly independent propositions. The two are one and the same. This is in fact an equivalent restatement of the Completeness theorem (which holds for contexts involving most common mathematical theories such as ZFC and Peano Arithmetic). In fact in such contexts, statements such as Con(PA) which is informally "the axioms of Peano Arithmetic are consistent" are in fact independent propositions and there are models of Peano Arithmetic that satisfy both Con(PA) and Not(Con(PA)).
What about contexts where the Completeness theorem fails (e.g. the second-order equivalents of those theories)? For example, what about CH in second-order ZFC (which is categorical under full semantics, i.e. has no independent statements and only undecidable ones?). In those contexts independence gets weird, especially in the way I think you're using it here. Usually we say that if a sentence s is independent of theory T then both s and Not(s) are consistent with T. However, consistency is phrased in terms of proofs. That is s is consistent with T if there is no finite proof in T of Not(s).
So if there are statements that are undecidable, but not independent (on the other hand all independent statements must be undecidable, although their independence may itself be decidable), we are effectively saying there exist consistent theories which have no mathematical model that satisfies them.
But this is a very weird way of doing mathematics (which is why most logicians are wary of using systems where the Completeness theorem fails to hold as foundations for mathematics). In particular it means that simply showing that a list of axioms is consistent is insufficient to allow its use. You must appeal to some other force to show that your mathematical theory is "valid" and allowed to be used.
But that's not a bar that modern mathematics tries to clear. Modern mathematicians do not try to justify the "validity" of mathematical theories beyond showing (or assuming) consistency.
For example, mathematicians generally don't hem and haw over whether the complex numbers "exist" in some real sense. They just show that they are a consistent extension of the real numbers and move on and use them.
To take a step back, the assertion that there are fundamentally undecidable mathematical propositions in the real sense you're laying out (as opposed to more formal treatments of truth in model theory) is a statement of the Platonist school of mathematical philosophy. Which is totally fine, but it is a statement of philosophy at the end of the day, rather than one of logic.
> To take a step back, the assertion that there are fundamentally undecidable mathematical propositions in the real sense you're laying out (as opposed to more formal treatments of truth in model theory) is a statement of the Platonist school of mathematical philosophy.
To go one step forward, these undecidable/uncomputable statements are computed by humans all the time, which makes the Platonist school of mathematics much more than an intellectual curio.
> these undecidable/uncomputable statements are computed by humans all the time
This is a far stronger statement than I think even Platonists would accept. What examples of non-computable statements that are computed by humans do you have in mind?
Even putting on my Platonist hat (which I'll wear for the rest of this reply), I would still argue, in line with xyzzyz's comment, that at most humans deal with computable descriptions of non-computable things. For example, the description of the real numbers is computable even if the real numbers themselves are not. Likewise even if the real numbers are not computable, we can only ever work with computably-definable (indeed only finitely definable) real numbers rather than arbitrary real numbers.
To use the map-territory analogy, while the territory is the thing we care about, the only things we can work with, manipulate, and compute are computable maps.
For example Chaitin's Constant is a non-computable real number. However, we never work with Chaitin's Constant directly, but rather with our finite description of it, from which we deduce all sorts of other facts.
There are therefore many things about Chaitin's Constant we will never know. But that is the Platonist's lot in life.
As an aside, there's in fact a formalization of this that states that a mathematical universe consisting solely of finitely definable entities and nothing else is perfectly reconcilable with ZFC. See https://mathoverflow.net/questions/44102/is-the-analysis-as-...). Now as I continue to wear my Platonist hat I don't believe for a second that this is in fact how the mathematical universe actually operates, but it is a good explanation for why it is beyond the mathematical abilities of humans to distinguish between this definable universe and the real universe we live in.
I'm not quite sure what you mean by undecidable propositions, but I assume it to mean something like the following?
Propositions which must be true or must be false (i.e. are satisfied/not satisfied respectively by all models which satisfy the overall theory), but for which there is no finite proof that they hold.
In contexts where Godel's Completeness theorem (not incompleteness theorem) holds undecidable propositions are exactly independent propositions. The two are one and the same. This is in fact an equivalent restatement of the Completeness theorem (which holds for contexts involving most common mathematical theories such as ZFC and Peano Arithmetic). In fact in such contexts, statements such as Con(PA) which is informally "the axioms of Peano Arithmetic are consistent" are in fact independent propositions and there are models of Peano Arithmetic that satisfy both Con(PA) and Not(Con(PA)).
What about contexts where the Completeness theorem fails (e.g. the second-order equivalents of those theories)? For example, what about CH in second-order ZFC (which is categorical under full semantics, i.e. has no independent statements and only undecidable ones?). In those contexts independence gets weird, especially in the way I think you're using it here. Usually we say that if a sentence s is independent of theory T then both s and Not(s) are consistent with T. However, consistency is phrased in terms of proofs. That is s is consistent with T if there is no finite proof in T of Not(s).
So if there are statements that are undecidable, but not independent (on the other hand all independent statements must be undecidable, although their independence may itself be decidable), we are effectively saying there exist consistent theories which have no mathematical model that satisfies them.
But this is a very weird way of doing mathematics (which is why most logicians are wary of using systems where the Completeness theorem fails to hold as foundations for mathematics). In particular it means that simply showing that a list of axioms is consistent is insufficient to allow its use. You must appeal to some other force to show that your mathematical theory is "valid" and allowed to be used.
But that's not a bar that modern mathematics tries to clear. Modern mathematicians do not try to justify the "validity" of mathematical theories beyond showing (or assuming) consistency.
For example, mathematicians generally don't hem and haw over whether the complex numbers "exist" in some real sense. They just show that they are a consistent extension of the real numbers and move on and use them.
To take a step back, the assertion that there are fundamentally undecidable mathematical propositions in the real sense you're laying out (as opposed to more formal treatments of truth in model theory) is a statement of the Platonist school of mathematical philosophy. Which is totally fine, but it is a statement of philosophy at the end of the day, rather than one of logic.