Kurpaska, Sławomir and Tyszka, Apoloniusz (2020) The physical impossibility of machine computations on sufficiently large integers inspires an open problem that concerns abstract computable sets X⊆N and cannot be formalized in the set theory ZFC as it refers to our current knowledge on X. [Preprint]

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Abstract

Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let β=(((24!)!)!)!, and let Φ denote the implication: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆(-∞,β]. We heuristically justify the statement Φ without invoking Landau's conjecture. The set X = {k∈N: (β<k) ⇒ (β,k)∩P(n^2+1) ≠ ∅} satisfies conditions (1)-(4). (1) There are a large number of elements of X and it is conjectured that X is infinite. (2) No known algorithm decides the finiteness/infiniteness of X . (3) There is a known algorithm that for every n∈N decides whether or not n ∈ X. (4) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n]. (5) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n] and some known definition of X is much simpler than every known definition of X\(-∞,n]. The following problem is open: Is there a set X⊆N that satisfies conditions (1)-(3) and (5)? The set X=P(n^2+1) satisfies conditions (1)-(3). Let [⋅] denote the integer part function. For every explicitly given integer m≥1, the set X={k∈N: [k/m]^2+1 is prime} contains m consecutive integers and satisfies conditions (1)--(3). The statement Φ implies that both sets X satisfy condition (5).

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Item Type:	Preprint
Creators:	Creators Email ORCID Kurpaska, Sławomir rtkurpas@cyf-kr.edu.pl 0000-0003-1885-4568 Tyszka, Apoloniusz rttyszka@cyf-kr.edu.pl 0000-0002-2770-5495
Keywords:	complexity of a mathematical definition; computable set X⊆N; current knowledge on X; explicitly known integer n bounds X from above when X is finite; infiniteness of X remains conjectured; known algorithm for every n∈N decides whether or not n∈X; large number of elements of X; mathematical statement that cannot be formalized in the set theory ZFC; no known algorithm decides the finiteness/infiniteness of X; physical impossibility of machine computations on sufficiently large integers
Subjects:	Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic
Depositing User:	Apoloniusz Tyszka
Date Deposited:	26 Jul 2020 14:37
Last Modified:	26 Jul 2020 14:37
Item ID:	17639
Subjects:	Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic
Date:	16 June 2020
URI:	https://philsci-archive.pitt.edu/id/eprint/17639

Available Versions of this Item

• The physical impossibility of machine computations on sufficiently large integers inspires an open problem that concerns abstract computable sets X⊆N and cannot be formalized in the set theory ZFC as it refers to our current knowledge on X. (deposited 20 Jun 2020 18:19)
  • The physical impossibility of machine computations on sufficiently large integers inspires an open problem that concerns abstract computable sets X⊆N and cannot be formalized in the set theory ZFC as it refers to our current knowledge on X. (deposited 26 Jul 2020 14:37) [Currently Displayed]
    • The physical impossibility of machine computations on sufficiently large integers inspires an open problem that concerns abstract computable sets X⊆N and cannot be formalized in ZFC as it refers to our current knowledge on X. (deposited 31 Aug 2020 23:21)
      • The physical limits of computation inspire an open problem that concerns abstract computable sets X⊆N and cannot be formalized in the set theory ZFC as it refers to our current knowledge on X. (deposited 16 Sep 2020 17:44)
        • The physical limits of computation inspire an open problem that concerns abstract computable sets X⊆N and cannot be formalized in the set theory ZFC as it refers to our current knowledge on X. (deposited 09 Oct 2020 17:41)

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