The physical limits of computation inspire an open problem that concerns decidable sets X⊆N and cannot be formalized in ZFC as it refers to the current knowledge on X

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Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,f(7)]. Let B denote the system of equations: {x_i!=x_k: i,k∈{1,...,9}} ∪ {x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. We write some system U⊆B of 9 equations which has exactly two solutions in positive integers x_9,...,x_9, namely (1,...,1) and (f(1),...,f(9)). No known system S⊆B with a finite number of solutions in positive integers x_1, . . . , x_9 has a solution (x_1,. . .,x_9)∈(N\{0})^9 satisfying max(x_1,..., x_9)>f (9). We write some system A of 8 equations. Let Λ denote the statement: if the system A has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. This heuristically proves the statement Φ. This proof does not yield that card(P(n^2+1))=ω. Algorithms always terminate. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known to us). Assuming that the infiniteness of a set X⊆N is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. *** (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) No known algorithm with no input returns the logical value of the statement card(X)=ω. (4) There are many elements of X and it is conjectured that X is infinite. (5) X has the simplest definition among known sets Y⊆N with the same set of known elements. *** Conditions (2)-(5) hold for X= P(n^2+1), condition (1) holds assuming the statement Λ. Conditions (1)-(4) hold for X={k∈N: (k>10^6) ⇒ (f(10^6),f(k))∩P(n^2+1)≠∅}, condition (5) fails as the set of known elements of X equals {0,...,10^6} . No set X⊆N will satisfy conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption.
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First archival date: 2017-10-17
Latest version: 227 (2021-07-01)
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