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Gödel's Incompleteness Theorems

Communications in Mathematical Analysis

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Intuitive Intro to Gödel's Incompleteness Theorem

What did Gödel show with his first incompleteness theorem?

One sufficient collection is the set of theorems of Robinson arithmetic Q. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems. The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms.

However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers.

Gödel's Incompleteness Theorems

A similar example is the theory of real closed fields , which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself in Tarski's formulation is an example of a complete, consistent, effectively axiomatized theory. The system of Presburger arithmetic consists of a set of axioms for the natural numbers with just the addition operation multiplication is omitted. In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results.

In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language this is sometimes called the principle of explosion , and is thus automatically complete.

A set of axioms that is both complete and consistent, however, proves a maximal set of non- contradictory theorems Hinman , p. The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent.

It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized. The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser using Rosser's trick. The resulting theorem incorporating Rosser's improvement may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated. First Incompleteness Theorem : "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.

The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in F. However, the sequence of steps is such that the constructed sentence turns out to be G F itself. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated.

Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation Smith , p. For this reason, the sentence G F is often said to be "true but unprovable. That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false.

The liar paradox is the sentence "This sentence is false.

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These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. That is, the system says that a number with property P exists while denying that it has any specific value.

For each formal system F containing basic arithmetic, it is possible to canonically define a formula Cons F expressing the consistency of F. This formula expresses the property that "there does not exist a natural number coding a formal derivation within the system F whose conclusion is a syntactic contradiction.

In the following statement, the term "formalized system" also includes an assumption that F is effectively axiomatized. Second Incompleteness Theorem : "Assume F is a consistent formalized system which contains elementary arithmetic. This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system.

The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the system F itself. There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of F as a formula in the language of F. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons F from the second incompleteness theorem is a particular expression of consistency.

Other formalizations of the claim that F is consistent may be inequivalent in F , and some may even be provable. For example, first-order Peano arithmetic PA can prove that "the largest consistent subset of PA" is consistent.

Communications in Mathematical Analysis

Perhaps an even cleaner example is Goodstein's theorem, due to Reuben Goodstein , which is purely number theoretic in nature. As we noted in Section 5. As axioms, it has the following seven assumptions:. We will now add two more examples of p. So B is infinite, but not enumerable. So U again is incomplete. Klein eds.

The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration. The standard proof of the second incompleteness theorem assumes that the provability predicate Prov A P satisfies the Hilbert—Bernays provability conditions. There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert—Bernays conditions.

Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic. This is because such a system F 1 can prove that if F 2 proves the consistency of F 1 , then F 1 is in fact consistent. For the claim that F 1 is consistent has form "for all numbers n , n has the decidable property of not being a code for a proof of contradiction in F 1 ". If F 1 were in fact inconsistent, then F 2 would prove for some n that n is the code of a contradiction in F 1. But if F 2 also proved that F 1 is consistent that is, that there is no such n , then it would itself be inconsistent.

This reasoning can be formalized in F 1 to show that if F 2 is consistent, then F 1 is consistent. Since, by second incompleteness theorem, F 1 does not prove its consistency, it cannot prove the consistency of F 2 either. This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic PA.

For example, the system of primitive recursive arithmetic PRA , which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually provide no interesting information if a system F proved its consistency. This is because inconsistent theories prove everything, including their consistency.

Thus a consistency proof of F in F would give us no clue as to whether F really is consistent; no doubts about the consistency of F would be resolved by such a consistency proof. The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that can be formalized in the system that is proved consistent. Gentzen's theorem spurred the development of ordinal analysis in proof theory. There are two distinct senses of the word "undecidable" in mathematics and computer science.

The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems , which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set see undecidable problem.

Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense. Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement.

Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the philosophy of mathematics. These results do not require the incompleteness theorem.

In , Saharon Shelah showed that the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory. Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound c such that no specific number can be proved in that system to have Kolmogorov complexity greater than c. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.

Intuitive Intro to Gödel's Incompleteness Theorem

In , Paris and Harrington proved that the Paris—Harrington principle , a version of the infinite Ramsey theorem , is undecidable in first-order Peano arithmetic , but can be proved in the stronger system of second-order arithmetic. Kirby and Paris later showed that Goodstein's theorem , a statement about sequences of natural numbers somewhat simpler than the Paris—Harrington principle, is also undecidable in Peano arithmetic.

Kruskal's tree theorem , which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem or its finite form is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more general graph minor theorem has consequences for computational complexity theory.

An introduction to mathematical theorems - Scott Kennedy

The incompleteness theorem is closely related to several results about undecidable sets in recursion theory. One such result shows that the halting problem is undecidable: there is no computer program that can correctly determine, given any program P as input, whether P eventually halts when run with a particular given input.

Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. You can find the take-home exam here. Please submit your answers by email. Thursdays, from 2pm to 4pm s.

What did Gödel show with his first incompleteness theorem?

Office hours by appointment, in room , Ludwigstrasse 31, second floor. You can find the book here. Here's a cool video. Very instructive! The result establishes that any theory extending elementary arithmetic is incomplete, this is, there will always be a sentence in the language of the theory that the theory neither proves nor refutes.

The exposition is carried out in two steps.

First, an introduction to first-order arithmetic is given. Students are presented with the following: first-order axiomatic theories, the language of first-order arithmetic, L , first-order theories of arithmetic Q Robinson Arithmetic and PA for "Peano Arithmetic" , the enumerability and denumerability of sets, the recursivity, semi-recursivity, definability in L , and representability in an axiomatic theory formulated in L, of a subset of natural numbers or a function from natural numbers to natural numbers, the standard model of L , arithmetical truth, the arithmetical hierarchy, and primitive recursive functions.

Exercises: about functions and enumerability except exercise 5. Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results. The formal explanations are interwoven with discussions of the wider significance of the two Theorems.

This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. Godel's Theorem in Philosophy of Mathematics. Edit this record.