They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent assuming it is indeed consistent. These results have had a great impact on the philosophy of mathematics and logic.
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They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent assuming it is indeed consistent.
These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial.
The present entry surveys the two incompleteness theorems and various issues surrounding them. These discoveries revolutionized the understanding of mathematics and logic, and had dramatic implications for the philosophy of mathematics. There have also been attempts to apply them in other fields of philosophy, but the legitimacy of many such applications is much more controversial.
Roughly, a formal system is a system of axioms equipped with rules of inference, which allow one to generate new theorems.
The set of axioms is required to be finite or at least decidable, i. The rules of inference of a formal system are also effective operations, such that it can always be mechanically decided whether one has a legitimate application of a rule of inference at hand.
Consequently, it is also possible to decide for any given finite sequence of formulas, whether it constitutes a genuine derivation, or a proof, in the system—given the axioms and the rules of inference of the system.
A formal system is complete if for every statement of the language of the system, either the statement or its negation can be derived i. A formal system is consistent if there is no statement such that the statement itself and its negation are both derivable in the system. Only consistent systems are of any interest in this context, for it is an elementary fact of logic that in an inconsistent formal system every statement is derivable, and consequently, such a system is trivially complete.
Accommodating an improvement due to J. Barkley Rosser in , the first theorem can be stated, roughly, as follows:. The sentence in question is a relatively simple statement of number theory, a purely universal arithmetical sentence. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another. For any statement A unprovable in a particular formal system F , there are, trivially, other formal systems in which A is provable take A as an axiom.
On the other hand, there is the extremely powerful standard axiom system of Zermelo-Fraenkel set theory denoted as ZF , or, with the axiom of choice, ZFC ; see the section on the axioms of ZFC in the entry on set theory , which is more than sufficient for the derivation of all ordinary mathematics.
Proving them would thus require a formal system that incorporates methods going beyond ZFC. A rough statement is:. In the case of the second theorem, F must contain a little bit more arithmetic than in the case of the first theorem, which holds under very weak conditions.
It is important to note that this result, like the first incompleteness theorem, is a theorem about formal provability, or derivability which is always relative to some formal system; in this case, to F itself. For many theories, this is perfectly possible. The existence of incomplete theories is hardly surprising. Take any theory, even a complete one see below for examples , and drop some axiom; unless the axiom is redundant, the resulting system is incomplete.
The incompleteness theorems, however, deal with a much more radical kind of incompleteness phenomenon. Unlike the above sort of trivially incomplete theories, which can be easily completed, there is no way of completing the relevant theories; all their extensions, inasmuch as they are still formal systems and hence axiomatizable, are also incomplete. They remain, so to speak, eternally incomplete and can never be completed. It is time to make this more precise.
The weakest standard system of arithmetic that is usually considered in connection with incompleteness and undecidability is so-called Robinson arithmetic due to Raphael M. Robinson; see Tarski, Mostowski and Robinson , standardly denoted as Q. As axioms, it has the following seven assumptions:. But it is a routine mechanical task to check whether a given sentence is an instance of this scheme. PA is generally taken as the standard first-order system of arithmetic.
It contains not just the above axioms of Q governing successor, addition and multiplication, but also defining axioms for all primitive recursive functions see the entry on recursive functions , and the application of the induction scheme is restricted to quantifier-free formulas i.
PRA , or something equivalent to it, is sufficient for developing the theory of syntax for formalized theories. It is often taken as the unproblematic background theory in which various other systems, whose legitimacy may be more controversial, are studied. A much stronger system than PA , important in the foundations of mathematics, which will be mentioned now and then below, is second-order arithmetic PA 2 also often denoted by Z 2.
It is more than sufficient for developing all ordinary analysis and algebra. Its language is a two-sorted first-order language see then entry on second-order and higher-order logic , i. PA 2 is a very strong theory. Obviously, it is assumed that our formal systems are also equipped with a system of rules of inferences and possibly some logical axioms , usually some standard system of classical logic though the incompleteness theorems do not essentially presuppose classical logic, but also apply to systems with, e.
The above standard systems all come with classical logic. For the first incompleteness theorem, Q is sufficient; for the standard proofs of the second theorem, something like PRA , at a minimum, is needed. Of course, there are many important and interesting theories in mathematics which are not even formulated in the language of arithmetic. However, the applicability of the incompleteness theorems can be dramatically extended outside the language of first-order arithmetic and its extensions, when it is noted that all that is needed is that weak theories such as Q , or PRA , can be interpreted in the system in question.
Most importantly, this involves various systems of set theory. For example, the incompleteness theorems hold for ZFC—Inf i. Roughly, a theory T 1 is interpretable in another theory T 2 if the primitive concepts and the range of the variables of T 1 are definable in T 2 so that it is possible to translate every theorem of T 1 into a theorem of T 2. One should not misunderstand such interpretations as providing anything like intuitive synonymity. Two theories may have radically different intended subject matter and yet, as formal systems, one may be interpretable in another.
As an illustration: a simple theory of ancestors may be, taken as a formal system, interpreted in arithmetic; obviously this does not mean that grandmothers and such are really numbers. What is significant is that interpretability preserves certain elementary formal properties of theories, most importantly, consistency: if T 1 is interpretable in T 2 and T 2 is consistent, T 1 is also consistent.
And any system in which Q can be interpreted is guaranteed to be essentially incomplete. However, for most purposes, it is just much simpler to establish the interpretability of Q in the theory at issue. In the case of the standard proofs of the second incompleteness theorem, substitute PRA for Q. On the other hand, not all theories of arithmetic are incomplete. These theories are, though, very weak. But in any case, at least a theory which deals with both addition and multiplication is needed.
More interestingly, the natural first-order theory of arithmetic of real numbers with both addition and multiplication , the so-called theory of real closed fields RCF , is both complete and decidable, as was shown by Tarski ; he also demonstrated that the first-order theory of Euclidean geometry is complete and decidable.
He also suggested, though did not demonstrate, that the proof could be adapted to apply also to the standard axiom systems of set theory such as ZFC.
What was still missing was an analysis of the intuitive notion of decidability, needed in the characterization of the notion of an arbitrary formal system. Recall that the set of axioms and the proof relation of a formalized system are required to be decidable. Mathematicians and logicians have implicitly used the intuitive notion of a decision method since antiquity, and as long as one asked for a positive solution, it was sufficient that one presented a concrete method that intuitively striked everyone as a mechanical method.
For the general limitative results, such as the general incompleteness theorems, or the undecidability results see 4. Instead of decidable sets or properties, one often considers effective or computable functions or operations, but in fact these are just two sides of the same coin—talk of one can be easily transcribed to talk of another. These proposals, though, all turned out to be equivalent. See the entries on computability , recursive functions and the Church-Turing thesis.
For a proper understanding of the incompleteness and undecidability results, it is vital to understand the difference between the two key notions regarding sets. In fact, this is, in the very abstract level, the essence of the first incompleteness theorem. However, if both a set and its complement are recursively enumerable, the set is recursive, i. In this section, the main lines of the proof of the first incompleteness theorem are sketched.
Naturally this implies normal consistency, and follows from the assumption that the natural numbers satisfy the axioms of F. From now on, it is assumed that the formalized systems under consideration contain Q and are assumed to be at least 1-consistent, unless otherwise stated. More precisely, two related notion are needed. A set S of natural numbers is strongly representable in F if there is a formula A x of the language of F with one free variable x such that for every natural number n :.
A set S of natural numbers is weakly representable in F if there is a formula A x of the language of F such that for every natural number n :.
It is obvious how all these notions are generalized to many-place relations. There are also related notions of representability for functions. As the incompleteness results in particular teach us, there are sets which are only weakly but not strongly representable the key example being the set of statements provable in the system. One should be careful here and focus on the relevant definitions, and not let the words mislead. Though these notions are relative to the formal system, it has turned out that strong and weak representability are extremely stable.
Quite independently of the particular formal system chosen, exactly the decidable, or recursive, sets relations are strongly representable, and exactly the semi-decidable, or recursively enumerable sets relations are weakly representable.
This holds for all formalized systems which contain Robinson arithmetic Q , from Robinson arithmetic itself to the strongest axioms systems of set theory like ZFC and beyond as long as they are recursively axiomatizable.
If the proofs of F are systematically generated, it will be eventually determined, for any given number n , whether it belongs to S or not—given that S is strongly representable in F. Both notions of representability—strong and weak—must be clearly distinguished from mere definability in the standard sense of the word.
There are many sets which can be defined in the language of arithmetic but not even weakly represented in any F , such as the set of consistent formulas, the set of sentences unprovable in the system F , or the set of Diophantine equations with no solutions see below. The essential point is that the chosen mapping is effective: it is always possible to pass, purely mechanically, from an expression to its code number, and from a number to the corresponding expression.
Today, when most of us are familiar with computers and the fact that so many things can be coded by zeros and ones, the possibility of such an arithmetization is hardly surprising. A little number theory then suffices to code sequences of numbers by single numbers.
Consequently, well-formed formulas, as sequences of primitive symbols, are each assigned a unique number. Finally derivations, or proofs, of the system, being sequences of formulas, are arithmetized, and are also assigned specific numbers. In this way, all the syntactic properties and operations can be simulated at the level of numbers, and moreover they are strongly representable in all theories which contain Q.
Let us denote the formula which strongly represents this relation in F itself as Prf F x , y. Let us abbreviate this formalized provability predicate as Prov F x. It follows that the latter is weakly representable though, it turns out, not strongly :.
Gödel's incompleteness theorems
This theorem applies to any formal theory that satisfies certain technical hypotheses, which are discussed as needed during the sketch. We will assume for the remainder of the article that a fixed theory satisfying these hypotheses has been selected. Throughout this article the word "number" refers to a natural number. The key property these numbers possess is that any natural number can be obtained by starting with the number 0 and adding 1 a finite number of times. Each formal theory has a signature that specifies the nonlogical symbols in the language of the theory. For simplicity, we will assume that the language of the theory is composed from the following collection of 15 and only 15 symbols:. This is the language of Peano arithmetic.
Proof sketch for Gödel's first incompleteness theorem
What is Gödel's proof?
Gödel’s Incompleteness Theorems