Statements 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
Agnieszka Kozdęba | Apoloniusz Tyszka
English. Edmund Landau’s conjecture states that the set Pn 2+1 of primes of the form n 2 + 1 is infinite. Landau’s conjecture implies the following unproven statement Φ: card(Pn 2+1 ) < ω ⇒ Pn 2+1 ⊆ [2, (((24!)!)!)!]. We heuristically justify the statement Φ. This justification does not yield the finiteness/infiniteness of Pn 2+1 . We present a new heuristic argument for the infiniteness of Pn 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 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.
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