The theory is that there are statements that are true but undecidable with any sufficiently strong decider. Those statements are proofs of that theory, they (or their negation) are true but undecidable.
You are right, in that those statements do satisfy the conclusion, and I don’t see why you’re being downvoted. The difference between them and the original, self referential, statement is generality. For example:
You: I know that ZF set theory is incomplete because the axiom of choice cannot be proven within it.
Some other guy: Okay then, how about we add the axiom of choice as another axiom, maybe this new system will be complete?
You: Nope, it still can’t prove or disprove the continuum hypothesis.
SUG: So I’ll add that in as another axiom, maybe that will finally patch it?
You: Nope, still doesn’t work because …
SUG: But if I add that as an axiom as well...
Hopefully you can see why this might keep going for quite a while, and given that its quite difficult to prove a statement is undecidable you will run out eventually. There’s nothing you can do to convince him those undecidable statements are symptoms of a general problem rather than just one-time flaws.
Compare to this:
Godel: ZF set theory is not complete, because I have this self-referential construction of an unprovable statement.
SUG: But what if I add your statement as another axiom?
Godel: Then my proof still applies to this new system you created, and I can construct another, similar statement.
SUG: But what if I...
Godel: There’s no point in you trying to continue this, whatever system you create my theorem will always apply.
The theory is that there are statements that are true but undecidable with any sufficiently strong decider. Those statements are proofs of that theory, they (or their negation) are true but undecidable.
You are right, in that those statements do satisfy the conclusion, and I don’t see why you’re being downvoted. The difference between them and the original, self referential, statement is generality. For example:
You: I know that ZF set theory is incomplete because the axiom of choice cannot be proven within it.
Some other guy: Okay then, how about we add the axiom of choice as another axiom, maybe this new system will be complete?
You: Nope, it still can’t prove or disprove the continuum hypothesis.
SUG: So I’ll add that in as another axiom, maybe that will finally patch it?
You: Nope, still doesn’t work because …
SUG: But if I add that as an axiom as well...
Hopefully you can see why this might keep going for quite a while, and given that its quite difficult to prove a statement is undecidable you will run out eventually. There’s nothing you can do to convince him those undecidable statements are symptoms of a general problem rather than just one-time flaws.
Compare to this:
Godel: ZF set theory is not complete, because I have this self-referential construction of an unprovable statement.
SUG: But what if I add your statement as another axiom?
Godel: Then my proof still applies to this new system you created, and I can construct another, similar statement.
SUG: But what if I...
Godel: There’s no point in you trying to continue this, whatever system you create my theorem will always apply.
SUG: Drat! Foiled again!
(Why is some other guy SUG and not SOG? Oversight? Euphony?)
Because I evidently can’t spell :(