Are infinite countable sets searchable in finite time? Let $S$ be a countable set with an infinite number of elements. To each element, one natural number is associated.
Is it possible to search $S$ for a random element (and retrieve its associated natural number) in a finite amount of time? Under which condition can this be achieved?
 A: Some very provisional and tentative thoughts...
As far as I understand things in Pure (Maths) Set Theory the concept of time plays no role by design, whereas in Applied (Maths) Set Theory time does play a role forged in the experience and observation of the real world. 
In regard to Applied (Maths) Set Theory, sets must be populated, enumerated and accessed in time, they cannot generally be existent a priori. The tasks of constructing and searching the set (whether completely synchronous or partially asynchronous) have to take place concurrently, to guarantee the search be completed in finite time. Even if considered notionally infinite in size (without a specified an upper bound to the number of elements it could contain), the set always remains partially constructed and finite in this process, whether the construction task proceeds in an ordered way or disordered way. 
In the case of certain simpler notionally infinite ordered sets where any arbitrary element value can be calculated very efficiently from the given element index; the element value calculation is by design one for one concurrent with each random search enquiry. The tasks of constructing and searching the set are fully synchronous and equally disordered in this case.
For set of Bell Numbers where every earlier Bell Number has to effectively be known to calculate the next, it is more efficient for the set construction task to proceed asynchronously to the search task.   (Calculating the n'th Bell Number using the first n Stirling Numbers of the Second Kind just leads to a more or less equivalent problem in an auxiliary set.)   
