gödel's incompleteness theorem explained

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Gödel’s Second Incompleteness Theorem Explained This is a followup to an earlier post where I talked about Gödel’s First Incompleteness Theorem . The First Incompleteness Theorem says that if a logical system is consistent, then Gödel’s statement (“This statement cannot be proven”) is true. More technically, the axioms should form a “first-order theory” with a “well-orderable language,” but there’s not need to go into details here. If there are numbers beyond the natural numbers, then the logical equivalency no longer holds. Here, I discuss the Second Incompleteness Theorem, and further implications. Technically, there is a particular statement of the consistency of the system that is unprovable. So if a proposition is unprovable, that means that it’s actually not contained by the theory, nor is the proposition’s negation. So, let g be the Gödel’s number of Gödel’s statement. But if we have an axiom saying Gödel’s statement is false, then this appears to require that the system is inconsistent. There are at least two distinct notions of consistency. ), This formulation is not Gödel’s original formulation, but it is very closely related. The formal system must include some amount of arithmetic, and must have a computable set of axioms. Each model is a bit different. I could say something like “All prime numbers are smaller than a billion.” It’s a false statement, but I can say it. This is called Gödel’s completeness theorem. I really liked the drawing of the non-standard car model. Could you remind me what the theorem was? That means that for any unprovable statement, there has to be a model of those axioms for which the statement is false. ( Log Out /  Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. Could I use it on an album cover or something? Of course, to “fix” this you could try to add that statement as an axiom.8, Then, since the statement is an axiom, it is trivially provable. Looking at the proof, the vaguest idea of an unprovable statment is “This statement is unprovable,” which seems… silly, and not worth your time. A model is an interpretation of those axioms. Basically, the axiom of induction says that proof by induction works. When people refer to “Goedel’s Theorem” (singular, not plural), they mean the incompleteness theorem that he proved and published in 1931. “(A implies B) or (not-A implies B)” is a tautology. We run into at least one of the following problems: 1. Of course, there are models that are very non-standard…. As long as your mathematics is complicated enough to include the natural numbers (which, I think we can agree, is not a particularly high bar), then it must have statements which cannot be proven true or false. Sure! More Voting Deadline Information, Courtesy of Snopes, Crip Dyke, Right Reverend Feminist FuckToy of Death & Her Handmaiden, Seth Meyers tries to keep up with fast-moving events, Gladiatorial games, now open to the public, Village of Bolingbrook employees scramble to prep the emergency pontoon boat for launch (Fiction), The October Crisis Remembered: Nous pardonnerons ou oublierons, I read popular physics: Interstellar Interlopers, Effective Altruism: an outsider perspective, Gay loneliness: critiques and counter-critiques. ω-consistency means that you also can’t prove a certain class of indirect contradictions. The issue is that now set theory cannot prove its own consistency. Gödel’s first incompleteness theorem says that if you have a consistent logical system (i.e., a set of axioms with no contradictions) in which you can do a certain amount of arithmetic4, then there are statements in that system which are unprovable using just that system’s axioms. –> Next Post: How Gödel Proved Math’s Inherent Limitations. Unfortunately, mathematicians have found statements of the kind that you might hope to prove, that are unprovable in standard mathematical systems. Doesn’t that contradict the axiom of induction? Your email address will not be published. Your email address will not be published. His basic procedure is as follows: Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable … In other words, as long as your logical system is complicated enough to include addition and multiplication, then your logical system is incomplete. You could say “For all predicates P, proof by induction works” (second-order Peano axioms). This result seems very promising for mathematics. Using the second-order axioms, my understanding is you can produce a formally consistent logic, but it does not correspond to any number system at all. Gödel’s completeness theorem implies that a statement is provable using a set of axioms if and only if that statement is true, for every model of the set of axioms. The Explosion Principle states that if you can prove a direct contradiction, then you can prove absolutely any statement. In other words, you can’t prove a statement both true and false. The second incompleteness theorem says that, within your mathematical system, you cannot prove that you can’t have contradictions. Change ), You are commenting using your Twitter account. ( Log Out /  Now suppose that the logical system can prove its own consistency. So if Gödel’s unprovable statement is false, that means there exists a proof of the unprovable statement. 0000000938 00000 n Now, using a more powerful system (one with more axioms), you can often prove the consistency (lack of contradictions) of a less powerful system (one with fewer axioms). However, since you can’t prove it true. This is a followup to an earlier post where I talked about Gödel’s First Incompleteness Theorem. Therefore, the logical system cannot prove its own consistency. It’s also a bit confusing because formally speaking, a theory (or what I was calling a “formal system”) is a collection of propositions, and each proposition contained is said to be proven by the theory. For instance, Peano arithmetic, which covers essentially the natural numbers and addition and multiplication, can be proven to be consistent with standard (ZFC) set theory, a more powerful system. But in both cases, the paradoxes and inability to prove a result are the result of picking the wrong axioms. 0000001089 00000 n Well, it’s not quite as bad as it sounds. If Gödel’s statement is false, isn’t the system inconsistent? Two years after Gödel published his completeness theorem, he published his incompleteness theorems. The theorem says that inside of a similar consistent logical system (one without contradictions), the consistency of the system itself is unprovable!5. @westlight snacker, <– Previous Post: Kurt Gödel’s Story And indeed, the non-standard number systems are inconsistent–in a certain sense. The normal natural numbers, follow these axioms, so are the standard model for them. Natural numbers are a model, and in this model, we have only the numbers we want to have, no more or less. You need a yet larger system to prove the consistency of set theory. With respect to the 1st incompleteness theorem, I believe that it would be more accurate to state that “when the system is consistent, the system necessarily comprises an undecidable proposition.”. It should be fun! We have some function Proof(x,y), which means that Statement x is a proof of Statement y. [ PnwC�n�W���[�7D�z�. -Comments RSS Yes, but then you could prove the system is inconsistent. 0000019563 00000 n Here, I discuss the Second Incompleteness Theorem, and further implications. How Gödel Proved Math’s Inherent Limitations, How Gödel Proved Math’s Inherent Limitations. There are things you can’t prove true or false! There can be several inequivalent such statements. Gödel’s statement implies “Gödel’s statement cannot be proven”, but the implication does not go the other way. But you can prove the consistency of a system by using a second, larger system. Gödel's first incompleteness theorem proves that "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In 1900, he gave a famous list of open problems in mathematics, the most important ones for the next century.

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