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sigh
I hate to break this to you, but: While both the Liar Paradox and Godel's Incompleteness Theorem rely on paradoxes generated by recursion, it does not follow that "every paradox generated by recursion is addressed by Godel's Theorem." Nor, as I posited, does Godel's Theorem indicate a method to assign a stable truth-value to those types of recursive paradoxes.
To wit: Godel's first theorem is that for any formal axiomatic system capable of generating arithmetic, there will always be a (recursive) theorem that cannot be proven within that formal axiomatic system. If your system excludes those types of statements (e.g. Principia Mathematica's meta-logical structure), then it is incomplete. If your system allows those types of statements, it will be inconsistent (as those G sentences cannot be consistent), which renders the formal system useless (via the principle of explosion).
Godel wasn't trying to legitimize recursive paradoxes. He was using them to show one of the limits of specific types of formal logical systems, such as the one Whitehead and Russel developed in an attempt to defend their logicism.
At best, the Liar Paradox is just an example of the type of recursive structure that Godel leveraged. He certainly wasn't saying "we can now say that the Liar Paradox is true."
Thus, a claim like "it is absolutely true that there are no absolute truths" is still self-defeating.
The only possible option is to declare it as a dialetheia, which generates its own raft of problems, including the fact that defending "it is absolutely true that there are no absolute truths" relies on far more than just one purported absolute truth, including "dialetheias exist" and "humans can properly identify dialetheias" and " 'it is absolutely true that there are no absolute truths' is in fact a dialetheia" and so on.
That's what I get for my wit being a tad too obtuse.