What does all this have to do with nonmathematical topics like religion, agnosticism, and metaphysics? The answer is that Goedel’s theorem points out a basic limitation of science. We notice that all of science taken as a whole is an example of an infinite mathematical system to which Goedel’s theorem does apply. The axioms of the system may be taken to be the “laws” or theories that have been discovered in the various disciplines. Giving credit to the scientists, let us assume that the laws discovered by them have been thoroughly examined so that they are not mutually contradictory. In other words, we are assuming that the set of axioms is not inconsistent. This is a statement in favor of science, because if the axioms are indeed inconsistent, then as it stands now, there is something wrong in science that needs to be rectified.
We can now apply Goedel’s theorem. We conclude that the set of laws that we have is incomplete, that there exist questions in the system that cannot be answered yes or no using these laws. The system under consideration is nothing but nature itself, so we conclude that the laws of science as they stand now cannot answer all questions about nature. Now, take a particular question. To answer it, we shall need to add a new axiom—that is, discover a new law. This particular question will now get answered, but science will now be a new system to which Goedel’s theorem shall again apply. Now there will be some other question that cannot be answered.
Notice that this process will never come to an end. Even if we worked for a million years, science at that time would still be an incomplete bunch of axioms, and there would be questions about nature that cannot be answered yes or no. We thus conclude that science has a basic limitation: that there will be no time in the future when it has completely fathomed the depths of nature. It is a set of axioms that will always remain incomplete. This is a fact that gives us a glimpse into reality.
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