What does $P$ in blackboard bold type of letter stand for? $\Bbb P$?

Question

In the first post of the thread "Cardinal number subtraction",

Cardinal number subtraction

there is a symbol for some kind of set which looks like this: $\Bbb P$

I am familiar with symbols for natural ($\mathbb{N}$), rational ($\mathbb{Q}$), real ($\mathbb{R}$), complex ($\mathbb{C}$) numbers, which are all written in blackboard bold type. I am not a mathematician, but I have encountered all kinds of mathematical symbols, but not this one. I am very curious about this symbol. Does it stand for something?

Alex

Answer 1

In the context of sets, if $X$ is a set the notation $\mathbb{P}(X)$ can be used to mean the power set of $X$. However, this is not universal notation, and the power set can also be referred to as $\mathcal{P}(X)$ or simply $P(X)$.

Answer 2

I have seen $\mathbb{P}$ used for primes and for irrationals. I believe, from the context of the question you mention, that it was primes. I would not recommend using it without defining it as the notation is not as standard as the notation you mention.

Answer 3

$\mathcal{P}(A)$ usually stands for the power set of a set A, meanwhile the set $\mathbb{P}(n)$ normally stands for the space of polynomials of order n

Answer 4

It could mean anything. From a partial order to the set of primes, to a probability function.

In the context of that question, I'm guessing it meant the set of primes, and the observation that $|\Bbb{P\setminus N}|=\aleph_0$ was supposed to be $|\Bbb{N\setminus P}|$ instead.

But it doesn't matter for the context of the question $\Bbb P$ can be any countable infinite set which contains infinitely many elements which are not in $\Bbb N$.