Statements and open problems on decidable sets 𝓧⊆ℕ that refer to the current knowledge on 𝓧
|Published in:||Issue 2, (Vol. 16) / 2022|
|Abstract.||Edmund Landau’s conjecture states that the set 𝓟𝒏𝟐+𝟏 of primes of the form 𝒏𝟐+𝟏 is infinite. Landau’s conjecture implies the following unproven statement 𝚽:𝐜𝒂𝒓𝒅(𝓟𝒏𝟐+𝟏) <𝝎⇒𝓟𝒏𝟐+𝟏⊆[𝟐,(((𝟐𝟒!)!)!)!]. We heuristically justify the statement 𝚽. This justification does not yield the finiteness/infiniteness of 𝓟𝒏𝟐+𝟏. We present a new heuristic argument for the infiniteness of 𝓟𝒏𝟐+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 on decidable sets 𝓧⊆ℕ that refer to the current knowledge on 𝓧.|
|Keywords:||Conjecturally Infinite Sets 𝓧⊆ℕ; Constructively Defined Integer 𝒏 Satisfies 𝐜𝒂𝒓𝒅(𝓧)<𝝎⇒𝓧⊆(−∞,𝒏]; Known Elements Of A Set 𝓧⊆ℕ Whose Infiniteness Is False Or Unproven; Mathematical Definitions, Statements And Open Problems With Epistemic And Informal Notions; Primes Of The Form 𝒏𝟐+𝟏; 𝓧 Is Decidable By A Constructively Defined Algorithm Which Is Currently Known.|
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