COMPLETE PROOFS OF GÖDEL’S INCOMPLETENESS THEOREMS. 3 hence these are recursive by P4. Notation. We write, for a ∈ ωn, f: ωn → ω a function. prove the first incompleteness theorem, and outline the proof of the second. (In fact, Gödel did not include a complete proof of his second theorem, but complete . The mathematician was Kurt Gödel, and the result proved in his paper became known as the Gödel Incompleteness Theorem, or more simply Gödel’s.
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The theorems are widely, but not universally, interpreted as showing that Hilbert’s program to find a complete and consistent set of axioms for all goedl is impossible. The first incompleteness theorem states theorej no consistent system of axioms whose theorems can be listed by an effective procedure i. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. They were followed by Tarski’s undefinability theorem on the formal undefinability of truth, Church ‘s proof that Hilbert’s Entscheidungsproblem is unsolvable, and Turing ‘s theorem that there is no algorithm to solve the halting problem. The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and effectively axiomatized, these concepts being detailed below.
Particularly in the context of first-order logicformal systems are also called formal theories. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation or rules of inference that allow for the derivation of new theorems from the axioms.
One example of such a system is first-order Peano arithmetica system in which all variables are intended to denote natural numbers. In other systems, such as set theoryonly some sentences of the formal system express statements about the natural numbers.
The incompleteness theorems are about formal provability within these systems, rather than about “provability” in an informal sense. There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.
This means that there is a computer program that, in principle, could enumerate all the theorrem of the system without listing any statements that are not theorems. The theory known as true arithmetic consists of all true statements about the standard integers in the language incompleheness Peano arithmetic. This theory is consistent, and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.
A set of axioms is syntacticallyor negation – complete if, for any statement in the axioms’ language, that statement or its negation is provable from the axioms Smithp.
It is not to be confused with semantic completeness, which means that the set of axioms proves all the semantic tautologies of the given language. But it is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the theorrm of logic alone.
In a mere system of logic it would be absurd to expect syntactic completeness. But in a system of mathematics, thinkers such tyeorem Hilbert had believed that it is just a matter of time to find such an axiomatization that would allow one to either prove or disprove by proving its negation each and every mathematical formula. A formal system might be syntactically incomplete by design, such as logics generally are.
Or it poof be incomplete simply because not all the necessary axioms have been discovered or prolf. For example, Euclidean geometry without the parallel postulate is incomplete, because some statements in the language such as the parallel postulate itself can goedl be proved from the remaining axioms. Similarly, the theory of dense linear orders is not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order.
In this case, there is no obvious candidate for a new axiom that resolves the issue. The theory of first-order Peano arithmetic is consistent, has incommpleteness infinite but recursively enumerable gosel of axioms, and can encode poof arithmetic for the hypotheses of the incompleteness theorem.
Thus, by the first incompleteness theorem, Peano Arithmetic is incompleteenss complete.
The theorem gives an explicit example of a statement of arithmetic that is neither provable nor disprovable in Peano’s arithmetics. Moreover, this statement is true in the usual model. Moreover, no effectively axiomatized, consistent extension of Peano arithmetic can be complete. A set of axioms is simply consistent if there is no statement such that incompletenexs the statement and its negation are provable from the axioms, and inconsistent otherwise.
Peano arithmetic is provably consistent from ZFC, but not from within itself.
Gödel’s incompleteness theorems
If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent. Additional examples of inconsistent theories arise from the paradoxes that result when the axiom schema of unrestricted comprehension is assumed in set theory.
The incompleteness theorems apply only to formal systems which profo able to prove a sufficient collection of facts about the natural numbers.
One sufficient gkdel 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. A similar example is the theory of real closed incompleteneesswhich 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.
Proof sketch for Gödel’s first incompleteness theorem
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 incompletdness 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 explosionand is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a maximal set of non- contradictory theorems Hinmanp. 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: If an axiom is ever added that makes the system complete, it does so at the cost of making the system gorel.
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.
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 gldel 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 Smithp. For this reason, the sentence G F is often said to be “true but unprovable.
That theorem shows that, when a sentence is uncompleteness 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 incompletdness. G says ” G is not provable in the system F. 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 gpdel 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 propf conclusion is a syntactic contradiction. In the following statement, the term “formalized system” also includes an assumption that F is effectively axiomatized.
Gödel’s incompleteness theorems – Wikipedia
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 Inconpleteness 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 tbeorem 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. Incpmpleteness formalizations of the claim that F is consistent may be inequivalent in Fand some may even be provable. Incompletenesd example, first-order Peano arithmetic PA can prove that “the largest consistent subset of PA” is consistent.
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 tueorem enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert—Bernays conditions.
Peano theorsm, 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 1then F 1 is in fact consistent.
For the claim that F 1 is consistent has form incomlpeteness all numbers nn 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 nthen it would itself be inconsistent. This reasoning can be incompleteness in F 1 incmpleteness 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, incopmleteness system of primitive recursive arithmetic PRAwhich 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 problemswhich 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.