This is a system I made, and I was wondering if someone had explored it before, and if it had a name.
You start with a list of natural numbers then remove every other number; I will call this first set $[0]$.
$[0]=0,2,4,6,8,10,12.. (2n)$
The list you started with now has every odd number left, so to find the next set you take every other number from the list.
$[1]=1,5,9,13,17,21..(1+4n)$
Your original list will now have $3,7,11,15...$ left. To get the next set you take every other number again.
$[2]=3,11,19,27..(3+8n)$
$[3]=7,23..(7+16n)$
etc..
Just like with modular arithmetic there are different variants of this system. The example I showed is analogous to mod2; I just call it type2 because you take every other number. In type3 you take every third number so that:
$[0]=0,3,6,9,12..(3n)$
You're left with $1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26,28,29,31...$ and every third number of that is:
$[1]=1,5,10,14,19,23,28,32$ (I'm not quite what the formula for these are.)
$[2]=2,8,16,22,29...$
etc..
I think I covered the idea. You could have type4, type5, type1, maybe try with negative numbers, etc.... I don't think I made any arbitrary decisions, except from starting the natural numbers list at $0$. Other than that, the system defined itself. I'm not sure how to write the definition mathematically though. Anyway, since the main idea is very basic, I'm sure this system has either been explored before, or its useless/boring. So I was wondering if anyone got any info on that, or maybe something to add?
Your process seems to have a $p$-adic number interpretation where $p=2$. In your given example, at each stage you eliminate half of the numbers that are left. This is very similar to the construction of the Cantor set also. In both of these spaces, the topology is totally disconnected. There are many variations of your process where certain classes of numbers or points are eliminated that are in certain open sets or intervals. The process seems a good way to construct unusual topological spaces.