zlacker

> Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.

Does this mean that the modern mathematician must be able to assemble all the digits of the number's representation (say, in decimal)? Given some formalization of this requirement, there would be a fairly set upper bound on the acceptable integers.

>>baddox+(OP)
It occurs to me that you could formalize the requirement by saying that there must be a provably terminating algorithm that produces precisely the decimal representation of the integer. That way, numbers like Graham's number can be submitted despite there being no physical way to represent the number in decimal (in the observable Universe).

>>baddox+81
Does the algorithm need to terminate? It seems like convergence would be sufficient. For example, we have no terminating algorithm to produce pi, but we have plenty of ways to precisly express it.

Of course, given the nature of this problem, you can define your converging algorithm to terminate when it is withing a certain error of what the "pure" answer would be. Of course, it is possible you would underestimate the number, and if someone could beet you by using the same method with a slightly different termination condition, so you may want to add 1...

>>gizmo6+J1
Good point, although this problem involves only integers, so I'm not sure if "convergence" makes sense. Pi, of course, is a computable number, implying that there is a terminating algorithm that, given input n, outputs the nth digit of pi's decimal representation.

>>baddox+(OP)
Determining something exactly does not mean providing its digits. It means uniquely identifying the semantic meaning of an expression. As an analogy, you can know what a word means when it's spoken to you even if you can't spell it. Or you can understand what a program written down on paper computes (and prove that it computes that thing) without knowing the explicit output of a particular input.

>>j2kun+kf
There must be more to it than that. For one thing, it would be necessary to determine which of the submitted numbers is larger, which according to my interpretation requires more than "uniquely identifying the semantic meaning of an expression."