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). The set X={k∈N: the number of digits of k belongs to P(n^2+1)} contains 10^{10^{450}} consecutive integers and satisfies conditions (1)(3). The statement Φ implies that both sets X satisfy condition (5).
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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:  20 Jun 2020 18:19  
Last Modified:  20 Jun 2020 18:19  
Item ID:  17352  
Subjects:  Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic 

Date:  21 April 2020  
URI:  http://philsciarchive.pitt.edu/id/eprint/17352 
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Open problems that concern computable sets X \subseteq N and cannot be formalized in ZFC as they refer to current knowledge about X. (deposited 21 Apr 2020 15:31)
 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) [Currently Displayed]
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