I’d maybe rather say that an infinite endeavor is one for which after any (finite) amount of progress, the amount of progress that could still be made is greater than the amount of progress that has been made, or maybe more precisely that at any point, the quantity of “genuine novelty/challenge” which remains to be met is greater than the quantity met already
[Edit: this line of reasoning is addressed later on; this comment is an artefact of not having first read the whole post, specifically see “With this picture in mind, one can see that whether an endeavor is infinite depends on one’s measure “] Imagine it makes sense to quantify the “genuine novelty” of puzzles using rational numbers, and imagine an infinite sequence of puzzles with “genuine novelty” of the i-th puzzle given {0..i-1} being equal to 1. Hence the sum of their “genuine novelty” diverges. Now, create a game which assigns 1/2^{i+1} points to solving the i-th puzzle. My guess as to the intended behaviour of your concept in this “tricky” example: - the endeavour of maximizing the score on this game is infinite. - the endeavour of being pretty sure to get a decent score at this game is not infinite. - the game itself is neither necessarily finite nor infinite. - for this game to support the weight of math/physics/cooking mushroom pies/etc..., it is not sufficient for me to become “very good at the game”. - The winner of this game is not determined by the finitude of the endeavors of the players [this is trivial, but note less trivially that for any finite number N, the prefix sequence maps to a finite endeavor, and the postfix sequence maps to an infinite endeavor with value in the game (if we let the endeavors “complete”) of 1-1/2^{N}, 1/2^{N} respectively].
A different way to make the “game” a test of the concept boundary would be: you get 1 point if you give the solution to at least one of the puzzles in the infinite sequence, else you get 0 points. This game lets you trivially construct infinite endeavors, despite yielding (for every infinite endeavor X) no benefit over a finite prefix (of X). Understanding all the different ways you could have won the game is an infinite endeavor.
[Edit: this line of reasoning is addressed later on; this comment is an artefact of not having first read the whole post, specifically see “With this picture in mind, one can see that whether an endeavor is infinite depends on one’s measure “]
Imagine it makes sense to quantify the “genuine novelty” of puzzles using rational numbers, and imagine an infinite sequence of puzzles with “genuine novelty” of the i-th puzzle given {0..i-1} being equal to 1. Hence the sum of their “genuine novelty” diverges. Now, create a game which assigns 1/2^{i+1} points to solving the i-th puzzle. My guess as to the intended behaviour of your concept in this “tricky” example:
- the endeavour of maximizing the score on this game is infinite.
- the endeavour of being pretty sure to get a decent score at this game is not infinite.
- the game itself is neither necessarily finite nor infinite.
- for this game to support the weight of math/physics/cooking mushroom pies/etc..., it is not sufficient for me to become “very good at the game”.
- The winner of this game is not determined by the finitude of the endeavors of the players [this is trivial, but note less trivially that for any finite number N, the prefix sequence maps to a finite endeavor, and the postfix sequence maps to an infinite endeavor with value in the game (if we let the endeavors “complete”) of 1-1/2^{N}, 1/2^{N} respectively].
A different way to make the “game” a test of the concept boundary would be: you get 1 point if you give the solution to at least one of the puzzles in the infinite sequence, else you get 0 points. This game lets you trivially construct infinite endeavors, despite yielding (for every infinite endeavor X) no benefit over a finite prefix (of X). Understanding all the different ways you could have won the game is an infinite endeavor.