In Mathematics, in Contradistinction to Any Empirical Science, the Predicate of the Current Knowledge Substantially Increases Its Constructive and Informal Part
Authors: Tyszka Apoloniusz 
Published: October 2023

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 theorem of any mathematician from past or present belongs to K. The set K exists only theoretically. Ignoring K and its subsets, sets exist formally in ZFC theory although their properties can be time-dependent (when they depend on K) or informal. In every branch of mathematics, the set of all knowable truths is the set of all theorems. This set exists independently of K. 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-trivial statements on decidable sets X&sube;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. The feature of mathematics from the article title is not quite new. Church's thesis, which is known from the beginning of computability theory, is based on the observation that the currently known computable functions are recursive, where the notion of a computable function is informal. It proves that the predicate of the current mathematical knowledge slightly increases the intuitive mathematics. For a set X&sube;N whose infiniteness is false or unproven, we define which elements of X are classified as known. No known set X&sube;N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. (1) A known algorithm with no input returns an integer n satisfying card(X)&lt;&omega; &rArr; X&sube;(-&infin;,n]. (2) A known algorithm for every k&isin;N decides whether or not k&isin;X. (3) No known algorithm with no input returns the logical value of the statement card(X)=&omega;. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y&sube;N with the same set of known elements. We prove that the set X={n&isin;N: the interval [-1,n] contains more than 29.5+(11!/(3n+1))∙sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. 501893&isin;X. Condition (1) holds with n=501893. card(X&cap;[0,501893])=159827. X&cap;[501894,&infin;)={n&isin;N: the interval [-1,n] contains at least 30 primes of the form k!+1}. If we add to X some set W satisfying 14&le;card(W)&le;23, then the following statements hold: X does not satisfy Condition (1), 159827+14&le;card(X), the above lower bound is currently the best known, card(X)&lt;&omega; &rArr; card(X)&le;159827+23, the above upper bound is currently the best known, X satisfies Conditions (2)-(5) except the requirement that X is naturally defined. Analogical statements hold, if we add to X the set ⋃_{x&isin;N, x divides 99!} {k&isin;N: k-501894 is the x-th element of P(n^2+1)}, where P(n^2+1) denote the set of primes of the form n^2+1. For a non-negative integer n, let &theta;(n) denote the largest integer divisor of 10^{10^{10}} smaller than n. Let &kappa;:N&rarr;N be defined by setting &kappa;(n) to be the exponent of 2 in the prime factorization of n+1. The set X={n&isin;N: (&theta;(n)+&kappa;(n))^2+1 is prime} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. Condition (1) holds with n=0^{10^{10}}. We present a table that shows satisfiable conjunctions of the form #(Condition1)&and; (Condition 2)&and;#(Condition 3)&and;(Condition 4)&and;#(Condition 5), where # denotes the negation &not; or the absence of any symbol. No set X&sube;N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption.