Undecidability and truth

Question

Are there undecidable problems for which a single truth exists?

For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds of geometries. That kind of proposition is not what I am talking about.

I know it's going to be a bit philosophical and maybe it doesn't have its place here but I would like an answer from a mathematical point of view.

So a bad example of what I am talking about : "is there a God?" this problem is undecidable however there is certainly only a single valid answer.

So I am not asking whether there is a God or not. I am asking if there are, in the field of mathematics, such questions known as undecidable, which only have a single valid answer (that will always remain a mystery, consequently).

Answer 1

There are many undecidable statements about the integers, but most people (including me) believe that the integers truly exist.

So any statement about the integers is either true or false, even if it is undecidable.

Answer 2

The answer to your question depends on what exactly you mean by "question", "decidability" and "only a single valid answer".

A question in the sense of computability theory is usually identified with the set of its yes-instances. E.g. the question "which natural numbers have the property P?" is identified with the set of natural numbers having the property P. Such a question (i.e. set of yes-instances) is then called decidable if there is an algorithm which given an (encoding of an) arbitrary instance decides in a finite amount of time whether this instance belongs to the set of yes-instances. In the example, the algorithm would receive a natural number n as input and decide in a finite amount of time whether the number n belongs to the set of yes-instances of the question P, i.e., whether n has the property P.

In this sense of decidability there are no undecidable questions with a "single valid answer": if there is only one answer, it is independent of the input. Hence the trivial algorithm which simply ignores its input and outputs "yes" (or "no", depending on what the valid answer is) serves to give a decision procedure for the question.

We might not know which of the two possible algorithms is correct, but the question is still decidable in the technical sense.