Defining a sequence by recursively excluding elements. Is this a valid way of defining a sequence? If not, why?
For example, suppose I want to define the sequence of prime numbers in the following way: 


*

*Start with sequence $(a_n) = 2,3,4,5,6,7,8...$ 

*Let $(a_n)_0 = (a_n)$ 

*Let $(a_n)_1$ be the sequence that is identical to $(a_n)_0$, but every multiple of $a_1$ is excluded, except of $a_1$ itself: $(a_n)_1 = 2,3,5,7,9,11,13...$  

*Let $(a_n)_2$ be the sequence that is identical to $(a_n)_1$, but every multiple of $a_2$ is excluded, except of $a_2$ itself: $(a_n)_2 = 2,3,5,7,11,13,17...$ 

*... 

*$(a_n)_m$ is the sequence that is identical to the sequence $(a_n)_{m-1}$, but every multiple of $a_m$ is excluded, except of $a_m$ itself.

*Thus, the sequence of primes is $((a_n)_m)_{m \in \mathbb{N^0}}$
On the surface it seems fine to me. However, I couldn't find any examples of recursive exclusion, which made me doubt it.

For clarification, I don't mean to ask whether my notation is correct. I'm asking about the concept of definition by recursive exclusion.

Looks pretty good to me. Though you are basically defining the final sequence as a limit to the procedure.
You can think of this process as creating a sequence of sequences which converge to the sequence of all prime numbers in an intuitive way. Formally, you can show that as the process is repeated infinite times, you would indeed be left with only and all the prime numbers, i.e, an element will be in the sequence iff it is a prime number.
That being said, I am not sure if the convergence would hold for some the obvious metrics that I could think of. (For example, it does not converge with $d(x_i,y_i)=\sup_i|x_i-y_i|$)
