Article
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Preserved in Portico This version is not peer-reviewed
The Predicate of the Current Mathematical Knowledge Substantially Increases the Constructive and Informal Mathematics and Why It Cannot Be Adapted to Any Empirical Science
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)
Version 3 : Received: 14 August 2023 / Approved: 15 August 2023 / Online: 15 August 2023 (08:49:43 CEST)
Version 4 : Received: 29 August 2023 / Approved: 30 August 2023 / Online: 31 August 2023 (03:53:31 CEST)
Version 5 : Received: 5 September 2023 / Approved: 6 September 2023 / Online: 7 September 2023 (05:01:30 CEST)
Version 6 : Received: 20 September 2023 / Approved: 21 September 2023 / Online: 22 September 2023 (05:14:58 CEST)
Version 7 : Received: 9 October 2023 / Approved: 10 October 2023 / Online: 10 October 2023 (10:05:52 CEST)
Version 8 : Received: 17 October 2023 / Approved: 18 October 2023 / Online: 19 October 2023 (04:49:53 CEST)
Version 9 : Received: 4 December 2023 / Approved: 6 December 2023 / Online: 6 December 2023 (12:11:37 CET)
Version 10 : Received: 14 January 2024 / Approved: 16 January 2024 / Online: 16 January 2024 (06:42:23 CET)
Version 2 : Received: 2 August 2023 / Approved: 3 August 2023 / Online: 4 August 2023 (07:45:49 CEST)
Version 3 : Received: 14 August 2023 / Approved: 15 August 2023 / Online: 15 August 2023 (08:49:43 CEST)
Version 4 : Received: 29 August 2023 / Approved: 30 August 2023 / Online: 31 August 2023 (03:53:31 CEST)
Version 5 : Received: 5 September 2023 / Approved: 6 September 2023 / Online: 7 September 2023 (05:01:30 CEST)
Version 6 : Received: 20 September 2023 / Approved: 21 September 2023 / Online: 22 September 2023 (05:14:58 CEST)
Version 7 : Received: 9 October 2023 / Approved: 10 October 2023 / Online: 10 October 2023 (10:05:52 CEST)
Version 8 : Received: 17 October 2023 / Approved: 18 October 2023 / Online: 19 October 2023 (04:49:53 CEST)
Version 9 : Received: 4 December 2023 / Approved: 6 December 2023 / Online: 6 December 2023 (12:11:37 CET)
Version 10 : Received: 14 January 2024 / Approved: 16 January 2024 / Online: 16 January 2024 (06:42:23 CET)
A peer-reviewed article of this Preprint also exists.
Tyszka, A. Constructive Mathematics with the Predicate of the Current Mathematical Knowledge. SSRN Electronic Journal 2024, doi:10.2139/ssrn.4710446. Tyszka, A. Constructive Mathematics with the Predicate of the Current Mathematical Knowledge. SSRN Electronic Journal 2024, doi:10.2139/ssrn.4710446.
Abstract
We assume that the current mathematical knowledge K is a finite set of statements which is time-dependent. This set exists only theoretically. In every branch of mathematics, the set of all knowable truths is the set of all theorems. This set exists independently of K. 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 and informal 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. Some subsets of K lead our attention to concrete finite sets. Among them is the set {k∈N\{0}: it is proved that k!+1 is prime} on which is based our main statement.
Keywords
conjecturally infinite sets X N; constructive algorithms; current mathematical knowledge; informal notions; known algorithms; known elements of N
Subject
Computer Science and Mathematics, Logic
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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