The theory of the foundations of mathematics - 1870 to 1940

form of Gödel’s first theorem: Let P be a set of Gödel numbers of all the provable sentences. If the set P̃∗ is expressible in L and L is correct, then there is a true sentence of L not provable in L. Proof: (based on [84]) Suppose L is correct and P̃ ∗ is expressible in L by a predicate H with Gödel number h. Let G be the diagonalization of H (i.e. the sentence H(h)). We show that G is true but not provable in L. H expresses P̃ ∗ in L, i.e. H(n) is true ↔ n ∈ P̃ ∗ for all n ∈ N. In particular, H(h) is true ↔ h ∈ P̃ ∗. We have that h ∈ P̃ ∗ ↔ d(h) ∈ P̃ ↔ d(h) / ∈ P . But since h is the Gödel number of H and by the definition of d, d(h) is the Gödel number of H(h) and so d(h) ∈ P ↔ H(h) is provable in L and d(h) / ∈ P ↔ H(h) is not provable in L. Now we have: H(h) is true ↔ H(h) is not provable in L. This means that H(h) is either true and not provable in L or false but provable in L. The latter alternative violates the hypothesis that L is correct. Hence it must be thatH(h) is true but not provable in L. Note that in this proof we have not defined the set T by a model but determined the truth of G by a metamathematical argument just as we have seen in step 5 of section 8.1, that is nevertheless commonly accepted by all mathematicians. Note also that the proposition G corresponds to the propoThis assumption is for technical reasons that make the proof more simple; Gödel’s original numbering did not have this restriction. 8.2. FORMALLY: GÖDEL’S INCOMPLETENESS THEOREMS 129 sition G of point 3 of section 8.1, since H(h) is a proposition that expresses of itself that it is not provable. Theorem: If L is correct and if the set P̃ ∗ is expressible in L, then L is incomplete. Proof: A system L that is correct and for which the set P̃ ∗ is expressible in L contains a sentence G that is true but not provable or refutable (By the previous theorem and the assumption of correctness). Hence G is true, but undecidable in L, and hence also incomplete. That is where the name incompleteness theorem comes from. By this theorem, it follows immediately that if a system is consistent , and the set P̃ ∗ is expressible in that system (which we will later see is true for a system of basic arithmetic) then it is incomplete. Note that this is the statement A → G of point 8 in section 8.1. When we study a particular language L, such as a system containing Peano’s arithmetic or the system of Principia Mathematica, we have to verify the assumption that P̃ ∗ is expressible in L. We can do this by separately verifying the following conditions. G1 : For any set A expressible in L, the set A∗ is expressible in L. G2 : For any set A expressible in L, the set à is expressible in L. G3 : The set P is expressible in L. Theorem: G1 ∧G2 ∧G3 → P̃ ∗ is expressible in L. Proof: G1 and G2 imply that for any expressible set A, à ∗ is expressible in L. In particular we then have that if P is expressible in L (i.e G3 holds), P̃ ∗ is expressible in L. Before we prove a general form of Gödel’s second incompleteness theorem, we introduce some more definitions. A sentence En is a Gödel sentence for a set A of natural numbers if either En is true and its Gödel number lies in A, or En is false and its Gödel number lies outside A, i.e. En is a Gödel sentence for A if and only if En ∈ T ↔ n ∈ A. Diagonal Lemma: For any set A, if A∗ is expressible in L, then there is a Gödel sentence for A. 130 CHAPTER 8. GÖDEL Proof: Suppose H is a predicate that expresses A∗ in L; let h be its Gödel number. Then d(h) is the Gödel number of H(h). For any number n, H(n) is true ↔ n ∈ A∗, therefore, H(h) is true ↔ d(h) ∈ A, and since d(h) is the Gödel number of H(h), then H(h) is a Gödel sentence for A. Lemma: If L satisfies G1, then for any set A expressible in L, there is a Gödel sentence for A. Proof: L satisfies G1, thus for any expressible set A, A∗ is expressible in L. Now we can apply the previous lemma to conclude that there is a Gödel sentence for A. With the diagonal lemma we can also prove the first theorem as follows: Since P̃ ∗ is expressible in L, by the diagonal lemma, there is a Gödel sentence G for P̃ . A Gödel sentence for P̃ is a sentence which is (by the definition of a Gödel sentence) true if and only if it is not provable in L. So for any correct system L, a Gödel sentence for P̃ is a sentence which is true but not provable in L. 8.2.2 The impossibility of an ‘internal’ proof of consistency With the diagonal lemma we can also prove a general form of Gödel’s second theorem, that was first formulated in this form by the Polish mathematician Alfred Tarski. A general form of Gödel’s second theorem (by Tarski) 1. The set T̃ ∗ is not expressible in L 2. If condition G1 holds, then T̃ is not expressible in L 3. If conditions G1 and G2 both hold, then the set T is not expressible in L (i.e. for systems for which G1 and G2 hold, truth within the system is not definable within the system.) Proof: To begin with, there cannot possibly be a Gödel sentence for the set T̃ because such a sentence would be true if and only if its Gödel number was not the Gödel number of a true sentence, and this is absurd. 8.2. FORMALLY: GÖDEL’S INCOMPLETENESS THEOREMS 131 1. If T̃ ∗ were expressible in L, then by the diagonal lemma, there would be a Gödel sentence for the set T̃ , which we have just shown is impossible. Therefore, T̃ ∗ is not expressible in L. 2. Suppose condition G1 holds. Then if T̃ were expressible in L, the set T̃ ∗ would be expressible in L, violating (1). 3. If G2 also holds, then if T were expressible in L, then T̃ would also be expressible in L, violating (2). Now we have seen both theorems in a general form, we will consider particular mathematical languages, starting with first order arithmetic, which we can build on in section 8.3 to prove the incompleteness of systems based on Peano’s arithmetic and other systems. 8.2.3 Gödel numbering and a concrete proof of G1, G2 and G3 This section will be completed in a later version of this document. For the moment we refer to Gödel’s original work that can be found in [93]. 132 CHAPTER 8. GÖDEL 8.3 Gödel’s theorem and Peano Arithmetic The classification of the various modes of syllogisms, when they are exact, has little importance in mathematics. In the mathematical sciences are found numerous forms of reasoning irreducible to syllogisms. G. Peano in [68, page 379] There are various different incompleteness proofs of Peano Arithmetic (with and without exponentiation). We mention three of them. The simplest uses a truth set defined by Tarski and shows that every axiomatizable subsystem of N (the complete theory of arithmetic) is incomplete. This proof of Gödel’s first theorem however cannot be formalized in arithmetic (since the truth set is not expressible in arithmetic), and was based on the underlying assumption that Peano Arithmetic is correct, implying that every sentence provable in Peano Arithmetic is a true sentence. Gödel’s original incompleteness proof involves the much weaker assumption of ω-consistency. Definition of simple consistency: An axiomatic system A is simply consistent := no sentence is both provable and refutable in A Definition of ω-inconsistent: An axiomatic system A is ω-inconsistent := there is a predicate F (w) (in one free variable w) such that the sentence (∃w :: F (w)) is provable but all the sentences F (0), F (1), . . . are refutable Definition of ω-incomplete: An axiomatic system A is ω-incomplete := A is a simply consistent axiomatic system in which all Σ0-sentences are provable Gödel’s original proof was based on the assumption of ω-consistency and shows that every axiomatizable ω-consistent system in which all true Σ0sentences are provable is incomplete. This proof is of course formalizable in Peano Arithmetic (and this is necessary for Gödel’s second theorem) and also shows that any axiomatic system A that is simply consistent and in which all Σ0-sentences are provable, is ω-incomplete. The third proof (1936) is due to Rosser and uses the even weaker assumption of simple consistency. It is based on an axiomatic system by the American mathematician Raphael Robinson (1912-1995), that we refer to as R. It 8.3. GÖDEL’S THEOREM AND PEANO ARITHMETIC 133 shows that every axiomatizable simply consistent extension of R is incomplete, but thereto uses a more elaborate sentence than the Gödel sentence ‘G is undecidable’. We intend to include the three proofs in a later version of this document. They can be found in [84] but in a particular presentation that does not use the concept of a model for axiomatic systems, and that sometimes attaches different meanings to established definitions, nevertheless it contains in our opinion one of the best discussions of Gödel’s incompleteness theorems. In a later version of this document we will also show how, given the proof of incompleteness of Peano Arithmetic, Gödel’s theorems apply to Principia Mathematica. We quote K. Gödel on the first page of [27]: The most comprehensive formal systems that have been set up hitherto are the system of Principia Mathematica on the one hand and the Zermelo-Fraenkel axiom system of set theory (further developed by J. von Neumann) on the other. These two systems are so comprehensive that in them all methods of proof today used in mathematics are formalized, that is, reduced to a few axioms and rules of inference. One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms”. 134 CHAPTER 8. GÖDEL