Constructive Mathematics with the Predicate of the Current Mathematical Knowledge

Apoloniusz Tyszka

doi:10.20944/preprints202307.2063.v10

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preprints.org > computer science and mathematics > logic > doi: 10.20944/preprints202307.2063.v10

Preprint Article Version 10 Preserved in Portico This version is not peer-reviewed

Constructive Mathematics with the Predicate of the Current Mathematical Knowledge

Apoloniusz Tyszka

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)

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 from both formal and constructive mathematics, which is time-dependent and publicly available. Any formal theorem of any mathematician from past or present forever belongs to K. We assume that mathematical sets are atemporal entities. In this article, they exist formally in ZFC theory although their properties can be time-dependent (when they depend on K) or informal. Algorithms always terminate. 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-trivially true statements on decidable sets X ⊆ N that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X.

Keywords

constructive mathematics; constructively defined algorithms; current mathematical knowledge; informal mathematics; known algorithms

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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Comments (1)

Comment 1

Received: 16 January 2024
Commenter: Apoloniusz Tyszka
Commenter's Conflict of Interests: Author

Comment: I changed the title, the Abstract, and section 1.

An older version of the preprint appeared in

Asian Research Journal of Mathematics 19 (2023), no. 12, pp. 69-79.

http://doi.org/10.9734/arjom/2023/v19i12773

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