preprints.org > computer science and mathematics > logic > doi: 10.20944/preprints202307.2063.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 27 July 2023 / Approved: 28 July 2023 / Online: 31 July 2023 (10:54:27 CEST)
Version 2 : Received: 2 August 2023 / Approved: 3 August 2023 / Online: 4 August 2023 (07:45:49 CEST)
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Version 10 : Received: 14 January 2024 / Approved: 16 January 2024 / Online: 16 January 2024 (06:42:23 CET)

This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on decidable sets X⊆N, Pi Mu Epsilon J. 15 (2023), no. 8, 493-504. The main results were presented at the 25th Conference Applications of Logic in Philosophy and the Foundations of Mathematics, see http://applications-of-logic.uni.wroc.pl/XXV-Konferencja-Zastosowania-Logiki-w-Filozofii-i-Podstawach-Matematyki. We assume that the current mathematical knowledge is a finite set of statements which is time-dependent. Nicolas D. Goodman observed that epistemic notions increase the understanding of mathematics without changing its content. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivial statements on decidable sets X⊆N that belong to constructive mathematics and refer to the current mathematical knowledge on X. This and the next sentence justify the article title. For any empirical science, we can identify the current knowledge with that science because truths from the empirical sciences are not necessary truths but working models of truth from a particular context. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,(((24!)!)!)!]. We heuristically justify the statement Φ. This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ.

composite numbers of the form 2^(2^n)+1, constructive algorithms, current mathematical knowledge, decidable sets X⊆N, epistemic notions, informal notions, known algorithms, known elements of N, primes of the form n^2+1, primes of the form n!+1, primes of the form 2^(2^n)+1

Computer Science and Mathematics, Logic

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