Is the following a valid application of Gödel's incompleteness theorem?

1 points by rasmafazi 8 years ago | 1 comment

In 1931, Kurt Gödel published his incompleteness theorems. His version is a bit complicated. I will use a much simpler analogy here.

Imagine that you create a table in which you store all theorems along with their proof. This is possible, because, since Richard's paradox, we know that the set of all proofs is countable. If someone asks you "Is this theorem provable?", you look it up in the table. If you can find it, you return the proof stored along with it. If not, you just say that it is unprovable. We can now take a look at Gödel's example theorem "This statement is unprovable". Is this statement provable? No. We cannot store it in the table, along with a proof. Hence, since you cannot return the proof, you must say that it is unprovable. Then, is this statement true? Yes. Since you can indeed not return its proof by looking it up in the table, it is true.

Consequently, we have discovered an entire class of statements that are true but not provable.

In that sense, the question "Where can we find proof that God exists?", just ignores Gödel incompleteness theorems. The person insisting that there should be a proof for every true statement, is wrong.

Therefore, it would be absolutely no surprise that God truly exists but that there is simply no proof for that. There is absolutely nothing wrong with that position. Hence, the God of gaps theory is ignorant. It fails to acknowledge the existence of true but not provable statements.

al2o3cr 8 years ago |

A key thing that's missing here: the incompleteness theorem is a statement about provability within a specified system of axioms that apply to statements about natural numbers.

It's totally possible to prove a Gödel sentence for one model of arithmetic in another that's "more powerful"; at a minimum, one can construct an expanded system that explicitly includes the sentence as an axiom. That system, however, has its own Gödel sentence.

The natural numbers part is equally important - the statement "the set of all proofs is countable" is trivially false for a system based on the reals, for instance. (Construct all statements of the form "exists x such that x > y" for y in R, a one-to-one correspondence between true statements and an uncountable set)

Applying results like the incompleteness theorem to statements that are entirely outside of its intended domain of discourse isn't doing mathematics, it's just bullshitting.