Like many laypeople, I was exposed to Gödel’s incompleteness theorems some years ago by Douglas Hofstadter (Gödel, Escher, Bach: An Eternal Golden Braid), and Sir Roger Penrose (The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics). For an excellent concise account of this topic, check out this entry in Wikipedia.
Over the years there has been much written on the possible implications of Gödel’s work for larger questions relating to philosophy of mind and artificial intelligence. J.R.Lucas, Penrose, and others have argued that Gödel’s theorems support the view that human intelligence transcends what can be accomplished by computers. Numerous experts in logic and mathematics have entered the debate to deflate these attempts as misguided extrapolations from Gödel’s setting of formal logical systems. Given these responses, I concluded that the Penrose-type arguments are not ultimately successful.
However it seems that one modest, but still important philosophical insight comes from considering this topic. Gödel showed that a consistent formal system cannot be complete in the sense of being able to prove its own consistency. This dovetails with the common sense insight that one cannot have complete objective knowledge of a system of which one is a part. This is why the larger scientific program, based on a methodological simulation of objectivity, runs into limits when it comes to explaining the inherently subjective phenomenon of human consciousness (here is a paper by philosopher Haim Gaifman which makes a similar point).
We humans cannot get outside the world-system and look back at it. Therefore the effort of seeking full objective truth will never fully succeed. We do have true knowledge of the world, but it is grounded in our existence as an experiencing subject integrated into the system.