Theorems and open problems that concern decidable sets X⊆N and cannot be formalized in mathematics understood as an a priori science as they refer to the current knowledge on X

Agnieszka Kozdęba (1, 2) , Apoloniusz Tyszka (2)

1 Faculty of Mathematics and Computer Science of the Jagiellonian University
2 Université d'Agriculture Hugo Kollataj

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). 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, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** We define a set X⊆N which satisfies Conditions (1)-(5) except the requirement that X is naturally defined. Let P(n^2+1) denote the set of primes of the form n^2+1. We present a new heuristic argument for the infiniteness of P(n^2+1). Conditions (2)-(5) hold for X=P(n^2+1). We discuss a conjecture which implies the conjunction of Conditions (1)-(5) for X=P(n^2+1). 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. We present a table that shows satisfiable conjunctions consisting of Conditions (1)-(5) and their negations.

Mots clés: physical limits of computation; primes of the form n^2+1; known elements of a set X⊆N whose infiniteness is false or unproven; conjecturally infinite set X⊆N; constructively defined integer n satisfies: (X is finite) ⇒ X⊆(-∞; n]; current knowledge on a set X⊆N; distinction between existing algorithms and constructively defined algorithms which are currently known; X is decidable by a constructively defined algorithm

Domaines: Logique [math.LO]; Théorie des nombres [math.NT]

Agnieszka Kozdęba, Apoloniusz Tyszka. Theorems and open problems that concern decidable sets X⊆N and cannot be formalized in mathematics understood as an a priori science as they refer to the current knowledge on X. 2021. ⟨hal-01614087v24⟩