I haven't gotten to the point where I can do a succinct summary of Godel's contributions, so these are just some comments and applications by others.
Lennox in God's Undertaker p185 applies Godel's work to what he regards as the unwarranted optimism of Dawkins' about a neatly closed "theory of everything." Lennox's comment: "Dawkin's optimism has proved unrealistic. Some ugly mathematical facts get in the way in the shape of Kurt Godel's famous finding that our familiar arithmetic and other larger mathematical systems cannot prove their own consistency and must contain propositions that are undecidable - that is, that cannot either be proved or disproved by arithmetical means. To put it another way, in any finite axiomatic system that is strong enough to include basic arithmetic, there will always be true statements that cannot be proved. Mathematician Nigel Cutland points out that this has negative implications for the possibility of a unified scientific theory which, of course, would have to include arithmetic."
Lennox's reference for Cutland is "Science and Christian Belief" 3 (1), 33-55, April 1991. He also references Raymond Smullyan's book "Forever Undecided - a puzzle guide to Godel", Oxford, 1988, which he describes as "A wonderful, imaginative introduction to " Godel's ideas.
Hawking quote about Godel closing off the complete theory of everything.
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