Gödel's Incompleteness Theorem - question

Question

Could someone correct me on the below logic.

If a statement cannot be proved then I cannot find a contradiction to said statement. If I cannot find a contradiction to said statement then that statement must be true. Hence all unprovable statements must be true.

Can we not take the italic text as a definition of what it means for something to be true? Or in other words, would it makes sense for something without contradictions to be false? Are there examples of such constructs?

Answer 1

Could someone correct me on the below logic.

If a statement cannot be proved then I cannot find a contradiction to said statement. If I cannot find a contradiction to said statement then that statement must be true. Hence all unprovable statements must be true.

The logic of your argument is flawless: it's a valid categorical syllogism.

The assumptions of your argument, however, are flawed:

If a statement cannot be proved then I cannot find a contradiction to said statement.

False. Unless you're working with an inconsistent set of assumptions, you cannot prove $0 \neq 0$. But you can find (and prove) its contradictory statement $0 = 0$

If I cannot find a contradiction to said statement then that statement must be true.

Also False. By Godel's Incompleteness Theorem, if you're working in a 'strong enough' yet consistent system, there will be statements that are true but that cannot be proven ... most notably the system's Godel sentence that effectively says 'The Godel sentence for this system cannot be proven in this very system'. And note that its contradictory statement 'The Godel sentence for this system can be proven in this very system' is false, and cannot be proven either. So you cannot prove the contradictory of the negation of the Godel sentence ... but it is not a true statement.