>> e.g. consider primes, infinite but become less frequent as you go.
Here is an alternative model:
Consider you have a container of numbers, starting with a single number inside, 1. Each turn, you can add any number of pairs of two numbers inside it (with replacement, so you can add two 1's), to discover more numbers.
The first turn you can add two 1's to discover 2. The next turn (with 1, 2) you can discover 3, 4, the turn after that (1, 2, 3, 4) you can discover (5, 6, 7, 8). The number of numbers you can discover in any single turn will continue to increase as you increase your number of discoveries.
I am thinking the density increases because with more discoveries you can form more permutations of existing knowledge.
Consider you have a container of numbers, starting with a single number inside, 1. Each turn, you can add any number of pairs of two numbers inside it (with replacement, so you can add two 1's), to discover more numbers.
The first turn you can add two 1's to discover 2. The next turn (with 1, 2) you can discover 3, 4, the turn after that (1, 2, 3, 4) you can discover (5, 6, 7, 8). The number of numbers you can discover in any single turn will continue to increase as you increase your number of discoveries.
I am thinking the density increases because with more discoveries you can form more permutations of existing knowledge.