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# Statements and open problems on decidable sets that contain informal notions and refer to the current knowledge on X

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For a set whose infiniteness is false or unproven, we define which elements of are classified as known. No known set 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 satisfying
.
(2)~A~known algorithm for every \mbox{} decides whether or not .
(3)~No known algorithm with no input returns the logical value of the statement .
(4)~There are many elements of~ and it is conjectured, though so far unproven, that is infinite.
(5) is naturally defined. The infiniteness of is false or unproven.
has the simplest definition among known sets with the same set of known elements.
The set
satisfies Conditions~(1)-(5) except the requirement that is naturally defined. . Condition~{\tt (1)} holds with . . . We present a table that shows satisfiable conjunctions of the form , where denotes the negation or the absence of any symbol.
No set 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 ~second or less.