It obeys all the "laws" (OK vagaries) of English grammar. There is nothing in the rules of grammar that requires a statement in English to actually be self-consistent. The problem only surfaces once you attach the associated meanings to the various components.

I just picked a classic to start off my prior comment which morphed more into praise for GEB than Goedel's brain breaker.

I have dim memories of a cracking constructive argument starting off with some very basic axioms where one was written as S, and two as SS and so on, then it all went a bit mad but the genius of Hofstadter is to make all that impenetrable palava nigh on accessible to the layman (with a bit of effort from the reader).

The horrendous thing about Goedel is that the final flourish is clearly correct (I assume that responsible adults have filled in the formal bits, I'm sticking to the lies to children version). It is both terrifying and perhaps obvious at the same time. I can imagine the sense of dread when mighty edifices such as Herr Hilbert's suddenly looked a bit shady and then anger, followed by disbelief and finally acceptance as the big hitters really got to grips with it. The world hasn't come to an end because of Goedel but it certainly got a bit more interesting.

I think that it is almost comforting that we have a system that can be complicated enough be to capable of saying things about itself that can't be proven within itself. I think that there is a good chance that our system of mathematics will eventually become complicated enough and no more. Obviously there will always be spherical cows and some absolutely mad numbers and when I say eventually - that will take forever (nearly).

It is still a paradox, a two step one, that drags the whole system into being a paradox. The false statements that have a proof do already abstractly exist.

There is no paradox in the statement "This sentence has no proof."