» Statements and open problems on decidable sets X \subseteq N that contain informal notions and refer to the current knowledge on X

For a set ${\mathcal X} \subseteq \N$ whose infiniteness is false or unproven, we define which elements of ${\mathcal X}$ are classified as known. No known set ${\mathcal X} \subseteq \N$ satisfies Conditions~{\tt (1)-(4)} and is widely known in number theory or naturally defined, where this term has only informal meaning.
{\tt (1)}~A~known algorithm with no input returns an integer $n$ satisfying
${\rm card}({\mathcal X})<\omega \Rightarrow$ ${\mathcal X} \subseteq (-\infty,n]$ .
{\tt (2)}~A~known algorithm for every \mbox{ $k \in \N$ } decides whether or not $k \in {\mathcal X}$ .
{\tt (3)}~No known algorithm with no input returns the logical value of the statement ${\rm card}({\mathcal X})=\omega$ .
{\tt (4)}~There are many elements of~ ${\mathcal X}$ and it is conjectured, though so far unproven, that ${\mathcal X}$ is infinite.
{\tt (5)} ${\mathcal X}$ is naturally defined. The infiniteness of ${\mathcal X}$ is false or unproven.
${\mathcal X}$ has the simplest definition among known sets ${\mathcal Y} \subseteq \N$ with the same set of known elements.
The set ${\mathcal X}=\{n \in \N:$
$the~interval~[-1,n]~contains~more~than$
$29.5+\frac{\textstyle 11!}{\textstyle 3n+1} \cdot \sin(n)~primes~of~the~form~k!+1\}$
satisfies Conditions~{\tt (1)-(5)} except the requirement that ${\mathcal X}$ is naturally defined.
$501893 \in {\mathcal X}$ . Condition~{\tt (1)} holds with $n=501893$ . ${\rm card}({\mathcal X} \cap [0,501893])=159827$ .
${\mathcal X} \cap [501894,\infty)=\{n \in \N:$ $the$ $interval$ $[-1,n]$ $contains$ $at$ $least$ $30$ $primes$
$of$ $the$ $form$ $k!+1\}$ . We present a table that shows satisfiable conjunctions of the form $\#{\rm (Condition}~{\tt 1}{\rm )} \wedge {\rm (Condition}~{\tt 2}{\rm )} \wedge \#{\rm (Condition}~{\tt 3}{\rm)} \wedge {\rm (Condition}~{\tt 4}{\rm )} \wedge \#{\rm (Condition}~{\tt 5}{\rm )}$ , where $\#$ denotes the negation $\neg$ or the absence of any symbol. No set ${\mathcal X} \subseteq \N$ will satisfy Conditions~{\tt (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.