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.
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 :(