Goldbach's Conjecture — Towards the Inconsistency of Arithmetic

1 Goldbach's Conjecture — Towards the Inconsistency of Arithmetic Ralf Wüsthofen Abstract. This paper proves that ZFC and Peano arithmetic (PA) are inconsistent, the latter result being a corollary of the former. We introduce a strengthened form of the strong Goldbach conjecture and show that both the conjecture and its negation can be deduced. The contradiction is triggered by the conjunction of two properties of an infinite set, which we use to reformulate the conjecture. We apply elementary number theory, where the constructive role of prime numbers within the natural numbers is a key point. Notations. Let denote the natural numbers starting from 1, let a denote the natural numbers starting from a > 1 and let 3 denote the prime numbers starting from 3. We use the syntactic entailment symbol ⊢ to make statements of the form ⊢ P for some statement P, which means that there exists a proof of P. Let SSGB denote the following strengthened form of the strong Goldbach conjecture: Every even number greater than 6 is the sum of two distinct odd primes. Theorem. ZFC is inconsistent. More precisely, both SSGB and the negation SSGB are provable. [1], [2] Proof. We define the set Sg := { (pk, mk, qk) | k, m ; p, q 3, p < q; m = (p + q) / 2 }. Sg has the following two properties. First, the whole range of 3 can be expressed by the triple components of Sg (”covering”). We prove this by dividing it into the following three cases. (i) x 3 is prime. Then, x = pk with p 3, k = 1. (ii) x 3 is composite and not a power of 2. Then, x = pk with p 3, k ≠ 1. (iii) x 3 is a power of 2. Then, x = (p + q)k / 2 with p = 3, q = 5, k = (a power of 2). So we have (C) x 3 Ǝ (pk, mk, qk) Sg x = pk x = mk. 2 Second, all pairs (p, q) of distinct odd primes are used in the definition of the set Sg (“maximality”). So we have (M) p, q 3, p < q k (pk, mk, qk) Sg, where m = (p + q) / 2. SSGB is equivalent to saying that every integer greater than or equal to 4 is the arithmetic mean of two distinct odd primes. So, under the assumption SSGB there is an n 4 that is different from all m defined in Sg, whereas under the assumption SSGB there is no such n. The following steps are independent of the choice of n if there is more than one. For example, the minimal such n works. The property (C) implies that for the above n, every nk', k' , equals a component of some Sg triple. The property (M) excludes the possibility that n is the arithmetic mean of a pair of distinct odd primes not used in Sg. So, (M) excludes the possibility that the question of whether SSGB holds or not depends on whether (M) holds or not. The basic idea is now the following. Due to the properties (C) and (M), from the assumption SSGB it follows that the set Sg is the union of the following triples. (a) Sg triples of the form (pk = nk', mk, qk) with k = k', if n is prime (b) Sg triples of the form (pk = nk', mk, qk) with k ≠ k', if n is composite and not a power of 2 (c) Sg triples of the form (3k, 4k = nk', 5k), if n is a power of 2 (d) all other Sg triples of the form (pk = nk', mk, qk), (pk, mk = nk', qk) or (pk, mk, qk = nk') (e) Sg triples of the form (pk ≠ nk', mk ≠ nk', qk ≠ nk'). From the assumption SSGB it follows that the set Sg is the above union of triples, where n is replaced by any y 3. Therefore, the Sg triples are the same in the case where we assume that n exists (i.e., SSGB) and in the case where we assume that n does not exist (i.e., SSGB). This is contradicted by the fact that under the assumption SSGB the numbers m defined in Sg take all integer values ≥ 4 whereas under the assumption SSGB they don’t. 3 To formalize this idea, we proceed as follows. We split Sg into two complementary subsets in the following way. For any y 3, we write Sg = Sg+(y) ∪ Sg-(y), with Sg+(y) := { (pk, mk, qk) Sg | Ǝ k' pk = yk' mk = yk' qk = yk' } Sg-(y) := { (pk, mk, qk) Sg | k' pk ≠ yk' mk ≠ yk' qk ≠ yk' }. We define S1 := { (pk, mk, qk) Sg | SSGB } and S2 := { (pk, mk, qk) Sg | SSGB }. I.e., S1 = Sg if SSGB is true, and S1 = { } if SSGB is false and S2 = Sg if SSGB is true, and S2 = { } if SSGB is false. Then, since under both assumptions SSGB and SSGB the properties (C) and (M) hold, we obtain (1.1) ⊢ y 3 ( SSGB => S1 = Sg+(y) ∪ Sg-(y) ) (1.2) ⊢ ( SSGB => S2 = Sg+(n) ∪ Sg-(n) ). So, since Sg+(n) ∪ Sg-(n) is independent of n, (1.1') ⊢ y 3 ( SSGB => S1 = Sg+(y) ∪ Sg-(y) ) (1.2') ⊢ y 3 ( SSGB => S2 = Sg+(y) ∪ Sg-(y) ). 4 Now we use the following principle. If two sets of (possibly infinitely many) z-tuples are equal, then the sets of their corresponding i-th components are equal; 1 ≤ i ≤ z. For this we define M1 := { m | (p, m, q) S1 } and M2 := { m | (p, m, q) S2 }. Then, applying the principle above to the middle component of the triples (p, m, q), the fact that each of the implications in ( (1.1') (1.2') ) is proved implies by transitivity (2.1) ⊢ y 3 ( SSGB => M1 = { m | (p, m, q) Sg+(y) ∪ Sg-(y) } ) (2.2) ⊢ y 3 ( SSGB => M2 = { m | (p, m, q) Sg+(y) ∪ Sg-(y) } ). Now we make use of the following metamathematical analogue of the first-order logic rule x ( P(x) Q(x) ) <=> ( x P(x) ) ( x Q(x) ). Lemma. (Distribution of Universal Quantifier over Conjunction in Proofs) Let T be a formal theory, and let P(x) and Q(x) be formulas. Then, T ⊢ x ( P(x) Q(x) ) <=> ( T ⊢ x P(x) ) ( T ⊢ x Q(x) ). (The lemma holds in standard proof systems for first-order logic; see standard textbooks, e.g. [3]). Applying the lemma to ZFC, we obtain that ( (2.1) (2.2) ) is equivalent to (2) ⊢ y 3 ( ( SSGB => M1 = { m | (p, m, q) Sg+(y) ∪ Sg-(y) } ) ( SSGB => M2 = { m | (p, m, q) Sg+(y) ∪ Sg-(y) } ) ). 5 We define M := { m | (p, m, q) Sg }. Then, since for every y 3 Sg+(y) ∪ Sg-(y) equals Sg by definition, for every y 3 { m | (p, m, q) Sg+(y) ∪ Sg-(y) } equals M by definition. If SSGB is true, M is equal to 4, and if SSGB is false, M is equal to some non-empty proper subset U of 4. It follows that there is exactly one set X { 4, U } that { m | (p, m, q) Sg+(y) ∪ Sg-(y) } is equal to. Since for every y 3 { m | (p, m, q) Sg+(y) ∪ Sg-(y) } = X regardless of whether SSGB or SSGB holds, in (2) we can replace { m | (p, m, q) Sg+(y) ∪ Sg-(y) } by X. Then, since ⊢ distributes over conjunction, we obtain (3) Ǝ! X { 4, U } ( ⊢ ( SSGB => M1 = X ) ⊢ ( SSGB => M2 = X ) ). Since the statements ( SSGB => M1 = X ) and ( SSGB => M2 = X ) depend on X and since X is the unique element of { 4, U } such that these statements hold, we will make use of the following rule. Let P1(A) and P2(A) be statements that depend on a set A. Let A be the unique element of { B1, B2, ..., Bz } such that P1(A) and P2(A) hold. Then, ( Ǝ! A { B1, B2, ..., Bz } ( ⊢ P1(A) ⊢ P2(A) ) ) => ( ( ⊢ P1(B1) ⊢ P2(B1) ) ( ⊢ P1(B2) ⊢ P2(B2) ) ... ( ⊢ P1(Bz) ⊢ P2(Bz) ) ). We apply the above rule with P1(A) = ( SSGB => M1 = A ), P2(A) = ( SSGB => M2 = A ) z = 2, B1 = 4, B2 = U. Then, since the left-hand side of the rule is true, we obtain 6 (3.1) ( ⊢ ( SSGB => M1 = 4 ) ⊢ ( SSGB => M2 = 4 ) ) (3.2) ( ⊢ ( SSGB => M1 = U ) ⊢ ( SSGB => M2 = U ) ). This implies (4.1) ⊢ ( SSGB => M2 = 4 ) (4.2) ⊢ ( SSGB => M1 = U ). On the other hand, we have ⊢ ( SSGB => M = 4 ) and ⊢ ( SSGB => M = U ). Therefore, since ⊢ ( SSGB => M1 = M ), ⊢ ( SSGB => M2 = M ), ⊢ ( SSGB => M1 = { } ) and ⊢ ( SSGB => M2 = { } ), we get (5.1) ⊢ ( SSGB <=> M1 = 4 ) (5.2) ⊢ ( SSGB <=> M2 = U ). Because of ( (5.1) (5.2) ) and because ⊢ ( SSGB => M2 = { } ≠ 4 ) and ⊢ ( SSGB => M1 = { } ≠ U ), 7 we have ⊢ ( M2 ≠ 4 ) and ⊢ ( M1 ≠ U ). Therefore, using ( (5.1) (5.2) ), from ( (4.1) (4.2) ) we obtain (6.1) ⊢ ( SSGB => FALSE ) (6.2) ⊢ ( M1 = 4 => M1 = U ) and (7.1) ⊢ ( M2 = U => M2 = 4 ) (7.2) ⊢ ( SSGB => FALSE ). Since (6.2) and (7.1) are both false, we obtain ⊢ SSGB and ⊢ SSGB. □ Corollary. Peano arithmetic (PA) is inconsistent. Proof. The term Sg from the above inconsistency proof is not a standard part of PA, but it can easily be defined within PA. This also applies to all other sets used in that proof, since they are all based on Sg or on . Therefore, the corollary is proved in the same way by using the operator ⊢ in the system PA. □ 8 References [1] Warning Signs of a Possible Collapse of Contemporary Mathematics, by Edward Nelson (2006). https://web.math.princeton.edu/~nelson/papers/warn.pdf [2] The Consistency of Arithmetic, by Timothy Y. Chow (2018). https://arxiv.org/pdf/1807.05641 [3] Introduction to Metamathematics, by Stephen Cole Kleene (North-Holland, 1952).