How could it be possible for the Goldbach conjecture to be undecidable?

Question

The answer to the question "Could it be that Goldbach conjecture is undecidable?" claims that it is possible for something such as the Goldbach conjecture to be undecidable, meaning that assuming that it is true and assuming that it is false would both lead to no contradiction.

But if it is undecidable, then, if we assume that it is false, it would mean that exists an even number that cannot be written as the sum of two primes. If a natural number exists, then it can be written down using a finite number of digits (any natural number is definable). This means that that number exists whether or not the conjecture is true, so if we assumed that it was true, there would be a contradiction, so it therefore can't be undecidable.

What is the flaw in what I just said?

Answer 1

What is the flaw in what I just said?

Your argument correctly shows that the Goldbach conjecture cannot be false and undecidable. However, it can be true and undecidable.

This is equivalent to saying that it can be true in the standard model of, say, Peano arithmetic, but false in some nonstandard model: in a nonstandard model there may be a nonstandard counterexample to Goldbach's conjecture which is not an ordinary integer, and so which cannot be written down in the usual sense.

The reason it can be true and yet assuming it's false doesn't lead to a contradiction is that if it's undecidable, Peano arithmetic doesn't know it's true, so Peano arithmetic can't use the fact that it's true in a proof.

Answer 2

if we assume that it is false, it would mean that exists an even number that cannot be written as the sum of two primes. If a natural number exists, then it can be written down using a finite number of digits

Ah not quite. If the proof the negation of Goldbach's conjecture is constructive, and if all the assumptions of the proof are constructive, then what you say is true. Converting a nonconstructive proof into a constructive proof is easy, just convert every nonconstructive inference step into an axiom. But converting a nonconstructive set of axioms into a constructive set of axioms is not trivial. Here are some examples of what would have to be true about every single assumption in the proof for the result to be constructive:

• For every assumption of the form $A \lor B$, you must be able to actually compute which of $\{A,~B\}$ is true. As a corollary:
• For every predicate in the proof $P$, where the assumption $P(x) \lor \lnot P(x)$ is assumed, you must be able to actually compute which of $\{P(x),~\lnot P(x)\}$ is true
• Every assumption of the form $A(x) \implies B(x)$, whenever $A(x)$ is computable, $B(x)$ is also computable
• Every assumption of the form $A(x)$, $A(x)$ must be computable

And there's more. Even 1 assumption could make the axiom set non-constructive. Let $G(m)$ be the predicate that holds when $m$ is less than 3, odd, or the sum of 2 primes. What if the (dis)proof of Goldbach's conjecture defines a predicate:

$$Q(n) = \forall m > n ~:~ G(m)$$

and made the assumption $Q(x) \lor \lnot Q(x)$. Can you write a computer program that, inputting $x$, outputs true if $Q(x)$ and false otherwise? If not, then the assumptions are not constructive (wrt your computing ability, that's a separate consideration though), and then you can't necessarily infer that the digits of the Goldbach counterexample are computable.