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 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 following problem is open: Is there a set X⊆N that satisfies conditions (1)(5) below? (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 k∈N decides whether or not k∈X. (4) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(∞,n]. (5) X is simply defined and we know an algorithm such that for every input k∈N it returns the sentence "k∈X" or the sentence "k∉X" and every returned sentence is true when k is sufficiently large. The simplest (in the sense of a verbal description) known to us such algorithm may return a false sentence only if k is small. We prove: (i) the set X ={k∈N: (β<k) ⇒ (β,k)∩P(n^2+1) ≠ ∅} satisfies conditions (1)(4), (ii) the set X = P(n^2+1) satisfies conditions (1)(3) and (5,) (iii) the statement Φ implies that the set X= P(n^2+1) satisfies condition (4).
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Item Type:  Preprint  

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Keywords:  computable set X⊆N; current knowledge on X; explicitly known integer n; machine computations on large integers; mathematical statement about X that refers to the current knowledge on X; physical limits of computation  
Subjects:  Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic 

Depositing User:  Apoloniusz Tyszka  
Date Deposited:  31 Aug 2020 23:21  
Last Modified:  31 Aug 2020 23:21  
Item ID:  18027  
Subjects:  Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic 

Date:  16 June 2020  
URI:  http://philsciarchive.pitt.edu/id/eprint/18027 
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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)
 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) [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 the set theory ZFC as it refers to our current knowledge on X. (deposited 26 Jul 2020 14:37)
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