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I want to define a set of prime numbers up to index $m$ as follows $$P_m=\{p_i:p_1\leq p_i\leq p_m,i\in \mathbb N\}$$ Now, $m=1,2,3,...$ should define the cardinality of this set. I like this compactness but am concerned that the $i\in\mathbb N$ implies that indices across all natural numbers must be chosen rather than stopping at index $m$. I originally did this for more generality such that $m$ can go to infinity. Should I instead define a subset of $\mathbb N$?

Nicojwn
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  • Your notation is not clear. In writing about primes, it is common to let $p_k$ denote the $k^{th}$ prime. Thus $p_1=2$, $p_2=3$, and so on. If so, then your set $P_m$ is precisely the set of the first $m$ primes. Is that what you intended here? – lulu Feb 04 '24 at 18:38
  • I'd just write "Let $P_m$ be the set consisting of the first $m$ prime numbers" or "Let $P_m={p_1,\dots,p_m}$, where $p_i$ is the $i$th prime number". – Karl Feb 04 '24 at 18:47
  • You could also clarify your definition by giving an example, e.g. $P_3={2,3,5}$. – Karl Feb 04 '24 at 18:49
  • @lulu specifically I want $p_1$ to be any prime making $P_m$ a subset of all primes. – Nicojwn Feb 04 '24 at 19:29
  • Then your notation is not good. As you see from the comments, most readers will assume you meant for $p_i$ to denote the $i^{th}$ prime. – lulu Feb 04 '24 at 19:30
  • I can't guess what you are trying to define. Are you saying that you want a sequence of $m$, possibly not distinct primes ${\psi_i}_{i=1}^m$ such that $\psi_1$ is the minimum (possibly not unique) and $\psi_m$ is the maximum (again, possibly not unique) but no assumption on the order of the ones in between? That seems very unusual. – lulu Feb 04 '24 at 19:32
  • Keep in mind: While Brevity is certainly a worthy goal in writing mathematics, Clarity is more important. You don't want your readers to have to make a long series of guesses to sort out what you mean. Add some example, write it out in standard English, if possible. – lulu Feb 04 '24 at 19:35

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