It's not as bad as you make it out to be. Let's take this one as an example.
...snip...
But doesn't this support my argument?
You (again, this is what drkitten did!) attempt to resolve the issue by stepping out one level. Instead of trying to assign a truth value to the statement, you try to assign a truth value to a claim
about the statement.
So this solves one problem, in that at least you can assign a value of false to the new sentence.
But it doesn't solve the problem of it being different from all other well formed sentences, because if you do the same for other well formed sentences -- as you show! -- you can always find
at least one way to make a true claim about them. The only way to make a true claim about the claim about the paradox is to step back further and say something like "it is true that I can't find a way to make a true claim about the truth of the sentence 'this statement is false'," and so forth.
And really, my argument isn't that "this statement is false" is somehow a magical anomaly vortex singularity, my argument is just that "this statement is false" is fundamentally different from all other well formed sentences in human language. I say people never come to a conclusion about it, you say you give up and give a postmortem -- same thing. The point is, we can't figure it out
like we can figure out other sentences. It might as well be a foreign language because it doesn't lead anywhere.
The whole point, of course, is that I think "this statement is false" is a
human Godel sentence. I don't agree with Penrose and the like that humans can
always step outside a given system. I don't think we can step outside
our system.
Now we can, of course, simply make the human Godel sentence another axiom of an "improved" human system, and again for the next Godel sentence we make for the "improved" system, and so forth -- but is this really "understanding" each system? I claim not. I claim that process would be
identical to programming a machine to generate a Godel sentence for any system you give it -- a trivial thing to do. That isn't "stepping outside the system," as AkuMani says, it is simply generating a statement.
EDIT: I think Hofstadter says something like this in GEB, although I didn't understand what he really meant at the time I read it.