Predicate logic doesn't really "fail" in natural science. It just isn't called into the game very often.
I probably should have used better phrasing. I know there are narrowly focused models for natural and human systems.
The statement that formalism was proven unfit by Gödel's incompleteness theorem was not mine, it comes from the linked article. However, since I'm of the Platonist school, I do agree with the observation.
IMHO, Platonism should be the least effected by Gödel because, under Platonism, the math already exists, waiting discovery. To me, that inoculates mathematical discovery from nay-sayings of all stripes.
I guess mathematical "ghosts" are ok with me (LOL!)
Any time you want something from me, you need but ask. I consider your archives to be world treasures, for which we all ought to feel indebted.
On previous threads you have spoken with authority and clarity on the principles which support Artificial Intelligence and therefore I consider you an expert!
Amusingly enough, I hail from the Stanford/Berkeley nexus that produced so much of AI, and was eagerly immersed in the stuff when I was a sprout, I am one of the numerous disaffected engineers who have grown rather impatient with the "discipline".
Although we have garnered some useful and decorative tools from AI, notably the programming_language/formal_math LISP (wherein data and code are treated identically) and such mainstays of database search as tree pruning and LISPish associative array processors, many of us consider it an advanced exercise in unreadable, undebuggable coding practices ensrined as design principles--so I have been disconnected from the field for a very long time now.
Nothing like familiarity to breed contempt, I guess. In that regard, one is tempted to point to the closing remarks of Russell, Whitehead, and Hilbert, after spending lifetimes trying to fruitlessly resuscitate the grand crystal palace of a hopeful 17th century clockwork-universe project of establishing complete, perfect, formal mathematical closure.
As if in one voice, they acknowledged that it probably couldn't be done, and that was before Godel ever hit center stage. Even without Godel, the problems of type conflict ( for example, the set of all sets contains itself, which constitutes a type conflict), undetermination, and under-determination had already undermined the project hopelessly.