Net: Construction of edge cases may be the only general rule (abstract) for the demonstration of the edge case.
Godel is more important in that he is one of multiple thinkers that falsified the analytic project, and the promise of making philosophy into a science.
The subject deserves a long treatment, but the underlying issue is the limits of set logic vs operational logic. Praxiology is NOT axiomatic(deductive and justificationary) but operational(Constructive and falsificationary) – Mises and Rothbard were punching above their weight.
Robert P. Murphy @BobMurphyEcon
“Gödel’s Incompleteness. I report you decide”
It’s more correct to say that (a) Babbage failed to inspire an operationalist(computational) movement. It took war and Turing. (b) the set-theoretic model of mathematics was a failure, and caused the set-theoretic model of language (philosophy) that was also a failure.
In other words, the logical model of set theory itself is a failure. Information for deduction will emerge in any given system, meaning that the only method of falsifying it (proof) is by construction (reconstruction). Closure is only provided by demonstration in reality.
All this means is that any deduction depends on the available hierarchy of constant relations, so that in any system of increasing complexity there will emerge niche conditions for which the solution (proof) is only by construction.
This is the point of praxeology(operationalism) in both economics and law. The spectrum of decidability in cases (questions) varies from the easily decidable on general rules to the individual accounting of edge cases.
The difference between computer science and humans is that we must be programmed via negativa (law), and computers via positiva (algorithms) because we have a world model, incentives within it (complete), and can always choose. They don’t and can’t. (incomplete)
So this is better said as ‘the philosophical program like the theological program has failed, leaving only the scientific program, and operationalism. Or better, theology was loose wisdom literature, philosophy slightly better, empiricism much better, and operationalism the best.
This is the fundamental insight of the failed programs of Mises, Brouwer, Bridgman, and to some degree Hilbert: operationalism only succeeded in physics and common law. It failed in every other field. With psychology and sociology the most obvious. Leaving room for postmodernism.