» 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 ${\mathcal X} \subseteq {\mathbb N}$ whose infiniteness is false or unproven, we define which elements of ${\mathcal X}$ are classified as known. No known set ${\mathcal X} \subseteq {\mathbb N}$ 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
${\rm card}({\mathcal X})<\omega \Rightarrow$ ${\mathcal X} \subseteq (-\infty,n]$ .
(2)~A~known algorithm for every \mbox{ $k \in \N$ } decides whether or not $k \in {\mathcal X}$ .
(3)~No known algorithm with no input returns the logical value of the statement ${\rm card}({\mathcal X})=\omega$ .
(4)~There are many elements of~ ${\mathcal X}$ and it is conjectured, though so far unproven, that ${\mathcal X}$ is infinite.
(5) ${\mathcal X}$ is naturally defined. The infiniteness of ${\mathcal X}$ is false or unproven.
${\mathcal X}$ has the simplest definition among known sets ${\mathcal Y} \subseteq {\mathbb N}$ with the same set of known elements.
The set ${\mathcal X}=\{n \in {\mathbb N}:$ $the~interval~[-1,n]~contains~more~than$ $29.5+\frac{\textstyle 11!}{\textstyle 3n+1} \cdot \sin(n)~primes~of~the~form~k!+1\}$
satisfies Conditions~(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 ${\mathcal X} \subseteq {\mathbb N}$ 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.