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

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

• Thanks for the edit; now it looks better with blackboard bold type of letters. :-) Mar 29 '15 at 6:54

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.

• I agree - in the context of the linked question it should be primes Mar 29 '15 at 6:48
• A sort of tangential remark, if my interpretation is correct they should have written $\mathbb{N} \setminus \mathbb{P}$ instead of $\mathbb{P} \setminus \mathbb{N}$.
– user29123
Mar 29 '15 at 6:59
• Yep, that part confused me for a bit too Mar 29 '15 at 7:04

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 $\Bbb N$.

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 $P(X)$.

• In the post, it was written like this: Well, |ℙ|=ℵ0, and |ℙ∖ℕ|=ℵ0, so perhaps ℵ0−ℵ0=ℵ0? I am not sure if it can denote powerset, but perhaps the set of prime numbers, as it was mentioned by Disciple of Barney? Mar 29 '15 at 6:49
• @AleksandarKatanovic Yeah, I looked at the post, and I agree with DiscipleofBarney's answer. Mar 29 '15 at 6:50
• Script P stands for power sets but $\mathbb{P}(n)$ normally stands for the space of polynomials of order n Mar 29 '15 at 6:51
• @AleksandarKatanovic (Though my answer is still valid for general use of the notation $\mathbb{P}$) Mar 29 '15 at 6:52
• @Dan I think your statement would be reasonable as a separate answer if you wanted to post it. Mar 29 '15 at 6:53

$\mathcal{P}(A)$ 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