The predicate of the current mathematical knowledge substantially increases the constructive mathematics what is impossible for any empirical science

This is a shortened and revised version of the article: A. Tyszka, Statements and open problems on decidable sets X⊆N, Pi Mu Epsilon J. 15 (2023), no. 8, 493-504. The main results were presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, see http://applications-of-logic.uni.wroc.pl/XXV-Konferencja-Zastosowania-Logiki-w-Filozofii-i-Podstawach-Matematyki. We assume that the current mathematical knowledge is a finite set of statements which is time-dependent. In every branch of mathematics, the set of all knowable truths is the set of all theorems. This set exists independently of our current scientific knowledge. Nicolas D. Goodman observed in Synthese that epistemic notions increase the understanding of mathematics without changing its content. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivial statements on decidable sets X⊆N that belong to constructive mathematics and refer to the current mathematical knowledge on X. This and the next sentence justify the article title. For any empirical science, we can identify the current knowledge with that science because truths from the empirical sciences are not necessary truths but working models of truth from a particular context

Theorems and open problems that concern decidable sets X⊆N and cannot be formulated in the formal language of classical mathematics as they refer to the current knowledge on X

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. We present a new heuristic argument for the infiniteness of P(n^2+1). Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. We write some system A⊆B 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. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). Algorithms always terminate. The next theorems and open problems justify the title of the article and involve epistemic and informal notions. 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. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies the conjunction of Conditions (1)-(5) for X=P(n^2+1). We define a set X⊆N which satisfies Conditions (1)-(5) except the requirement that X is naturally defined. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. 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

The predicate of the current mathematical knowledge substantially increases the constructive mathematics what is impossible for the empirical sciences

This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on decidable sets X⊆N, Pi Mu Epsilon J. 15 (2023), no. 8, 493-504. The main results were presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, see http://applications-of-logic.uni.wroc.pl/XXV-Konferencja-Zastosowania-Logiki-w-Filozofii-i-Podstawach-Matematyki. We assume that the current mathematical knowledge is a finite set of statements which is time-dependent. Nicolas D. Goodman observed that epistemic notions increase the understanding of mathematics without changing its content. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivial statements on decidable sets X⊆N that belong to constructive mathematics and refer to the current mathematical knowledge on X. This and the next sentence justify the article title. For any empirical science, we can identify the current knowledge with that science because truths from the empirical sciences are not necessary truths but working models of truth from a particular context. 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,(((24!)!)!)!]. We heuristically justify the statement Φ. This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ

Statements and open problems on decidable sets X⊆N that contain informal notions and refer to the current knowledge on X

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))f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. We write some system A⊆B 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. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. Algorithms always terminate. The next statements and open problems justify the title of the article and involve epistemic and informal notions. 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. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies Condition (1) for X=P(n^2+1). We define a set X⊆N which satisfies Conditions (1)-(5) except the requirement that X is naturally defined. The conjecture that there are infinitely many primes of the form k!+1 implies that the set N \setminus X is finite. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. 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. The article was presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, http://www.applications-of-logic.uni.wroc.pl/Program-

The predicate of the current mathematical knowledge increases the scope of mathematics what distinguishes mathematics from other fields of study

This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on decidable sets X⊆N, Pi Mu Epsilon J. 15 (2023), no. 8, pp. 493-504. The main results were presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, see http://applications-of-logic.uni.wroc.pl/XXV-Konferencja-Zastosowania-Logiki-w-Filozofii-i-Podstawach-Matematyki. Nicolas D. Goodman observed that epistemic notions increase the understanding of mathematics without changing its content. We show that the predicate of the current mathematical knowledge increases the scope of mathematics. This distinguishes mathematics from other fields of study. We assume that the current mathematical knowledge is a finite set of statements, which is time-dependent. 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,(((24!)!)!)!]. We heuristically justify the statement Φ. This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. The distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known inspires statements on decidable sets X⊆N that refer to the current mathematical knowledge on X