When setting number-sequence puzzles, is there any accepted method to prevent arbitrary solutions?

When setting a number sequence puzzle, one normally gives a finite sample and asks for the rule that specifies the members.

The sample has to be finite because it's impossible to uniquely specify an infinite series without giving away the answer. (Note: Is this true?)

Of course the problem is that any finite string of numbers can have an infinite number of rules that specify it.

Example

What is the next number in this series?

2 3 4 7 9 ?

To which someone could say: These are members of the set {2 3 4 7 9 73} so the answer is 73.

or

This is the union of the sets {2, 3, 4, 7, 9} and {10, 11, 12, 13, ...} so the answer is 10.

This is taking things to an extreme but how is it possible to counter such suggestions?

Note: The real-life example that caused me to pose the question is here Can you generate the next number in this integer series and describe the rule? and the first answer proposed was some arbitrary union of sets.

Question

Are there any recognised ways, or can you suggest a way, of designing number sequence puzzles that can tie down the generating rule exactly without actually specifying the answer? If not, is the number sequence puzzle genre dead?