There was a lot of optimism at the start of the 20th century regarding the power of axiomatic systems (in particular mathematics) to explain the universe. An axiomatic system is one which does not rely on observation, induction, or anything silly like that - it starts with a small number of axioms which are assumed a priori to be true, and some rules of deduction. Then you can use the axioms and the rules in a flawlessly logical way to prove theorems of the system which are incontrovertibly true. For example, you can find the axioms of number theory here. If you were so inclined, you could use these to prove, for example, that there are infinitely many prime numbers. There is no way of arguing with this proof - it is incontrovertibly true, since all you have done is use logic to work on the axioms. People's optimism stemmed from the fact that they thought the incontrovertible axioms that founded these systems were not just plucked out of thin air, but actually corresponded to something incontrovertibly true about the universe; thus if you selected your system and your axioms properly, you could start to prove things not just about mathematics but also about astronomy, physics, chemistry, etc., in a rigorously logical and incontrovertible way. In other words, you could use pure logic to discover the truths of the universe. (People like Reimann came along and showed that the initial step of choosing axioms that correspond to incontrovertible properties of the real universe is not as easy as it sounds - but that's another story.)
People like Bertrand Russell reckoned that, assuming you'd set up your system properly (i.e. chosen sensible axioms and rules of deduction), you must be able come up with some sort of automated process that would simply apply the rules to the axioms and come out with true theorems. You can see that this is a realistic belief, certainly for the "first level" of theorems - those that you obtain simply by applying one of the rules of deduction to one of the axioms. Then you can just continue using your rules and axioms to work on these "first level" theorems to generate second and subsequent levels of theorems, all of which are guaranteed to be true. Thus - surely - we can reliably generate ALL of the true theorems of an axiomatic system. Perhaps there might be a problem with your automated process where it can get stuck in an infinite loop or something so it might be quite difficult to do this in practice (and of course it might take a very long time) but at the very least all of the true theorems must be "out there" somewhere, waiting for us to discover them.
Well, along came Godel, who wanted to challenge the concept of such "complete and consistent" systems ("complete" because the system contains all the true theorems - i.e. all of the true theorems are "out there" somewhere; and "consistent" because it doesn't have any theorems that are wrong or disagree with each other). He did the following in a very clever and rigorous way, but essentially what he said was this. Consider the theorem which says, "This theorem is not part of your complete and consistent system". There are two possibilities. First, that theorem is true. In that case, what it says is true: "This theorem is not part of your complete and consistent system". Thus we have identified a true theorem which is not part of the system, and so the system is not complete. On the other hand, suppose the theorem is false. Then what it says is false. In other words, in fact it IS part of our supposedly complete and consistent system. In that case we have identified a false theorem which is part of our system, and so the system is not consistent. You can apply this procedure to any axiomatic system with rules of inference. Godel thus proved, in a rigorously logical way, that it is not possible to formulate any axiomatic system which is both complete and consistent. We are never going to discover all the truths about anything at all by using logic alone.
Everyone was extremely disappointed.
People like Bertrand Russell reckoned that, assuming you'd set up your system properly (i.e. chosen sensible axioms and rules of deduction), you must be able come up with some sort of automated process that would simply apply the rules to the axioms and come out with true theorems. You can see that this is a realistic belief, certainly for the "first level" of theorems - those that you obtain simply by applying one of the rules of deduction to one of the axioms. Then you can just continue using your rules and axioms to work on these "first level" theorems to generate second and subsequent levels of theorems, all of which are guaranteed to be true. Thus - surely - we can reliably generate ALL of the true theorems of an axiomatic system. Perhaps there might be a problem with your automated process where it can get stuck in an infinite loop or something so it might be quite difficult to do this in practice (and of course it might take a very long time) but at the very least all of the true theorems must be "out there" somewhere, waiting for us to discover them.
Well, along came Godel, who wanted to challenge the concept of such "complete and consistent" systems ("complete" because the system contains all the true theorems - i.e. all of the true theorems are "out there" somewhere; and "consistent" because it doesn't have any theorems that are wrong or disagree with each other). He did the following in a very clever and rigorous way, but essentially what he said was this. Consider the theorem which says, "This theorem is not part of your complete and consistent system". There are two possibilities. First, that theorem is true. In that case, what it says is true: "This theorem is not part of your complete and consistent system". Thus we have identified a true theorem which is not part of the system, and so the system is not complete. On the other hand, suppose the theorem is false. Then what it says is false. In other words, in fact it IS part of our supposedly complete and consistent system. In that case we have identified a false theorem which is part of our system, and so the system is not consistent. You can apply this procedure to any axiomatic system with rules of inference. Godel thus proved, in a rigorously logical way, that it is not possible to formulate any axiomatic system which is both complete and consistent. We are never going to discover all the truths about anything at all by using logic alone.
Everyone was extremely disappointed.
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