The predicate of the current mathematical knowledge substantially increases the constructive and informal mathematics and why it cannot be adapted to any empirical science

Apoloniusz Tyszka (1)

1 Université d'Agriculture Hugo Kollataj

The main mathematical results of this article 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 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.

Mots clés: conjecturally infinite sets X⊆N; constructive algorithms; current mathematical knowledge; informal notions; known algorithms; known elements of N

Domaines: Logique [math.LO]; Théorie des nombres [math.NT]

Apoloniusz Tyszka. The predicate of the current mathematical knowledge substantially increases the constructive and informal mathematics and why it cannot be adapted to any empirical science. 2023. ⟨hal-01614087v33⟩