What is the difference between natural numbers and positive integers? I was reading sets and came to some reserved letters for a few sets.
Two of them really confused me. They were -
$\mathbb N$ :  For the set of natural numbers.
$\mathbb Z^+$ : For the set if all positive integers.
In my sense, both the sets contain $\{1,2,3,\dots\}$
Then, why are they considered different?
I searched a little on this topic and got this, but it doesn't tell anything about significance of two different sets.
 A: The positive integers are $\mathbb Z^+=\{1,2,3,\dots\}$, and it's always like that.
The natural numbers have different definitions depending on the book, sometimes the natural numbers is just the postivite integers $\mathbb N=\mathbb Z^+$, but other times the natural numbers are actually the non-negative numbers $\mathbb N=\{0,1,2,\dots\}$.
Some people also write $\mathbb N_0=\{0,1,2,\dots\},\mathbb Z^+=\{1,2,3,\dots\}$ and completely avoid $\mathbb N$ due to this ambiguity.
If you want to be completely unambigiuous, you should use the words positive integers and nonnegative integers for these sets.
A: You should be aware that some authors define $\mathbb{N}$ to include zero.  This isn't of much consequence in itself since the properties of the set are preserved: there is a bijection between $\mathbb{N}$ with zero and $\mathbb{N}$ without zero, both are well-ordered, and so forth—effectively, we've done nothing but "relabel" the elements.  
Only when we start adding structure to these elements does the distinction become important.  For instance, if we define an addition $+: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, we might make $0$ an additive identity.  Therefore, when one writes "$\mathbb{N}$" in such a scenario (most scenarios), then it should be made clear which definition is intended.
Now, if we take both to mean the set $\{1, 2, 3, \cdots\}$, then whether one writes $\mathbb{N}$ or $\mathbb{Z}^+$ is immaterial.  However, using $\mathbb{Z}^+$ removes ambiguity since $\mathbb{Z}^+$ definitively does not include zero, and we would not have to go out of our way defining $\mathbb{N}$. 
A: Regarding the question of whether or not the natural numbers should include zero, there are two arguments in favor of doing so that I find compelling:
1) By including zero, the natural numbers can then be used to indicate cardinalities for all finite sets.  If zero is not included, then the cardinality of the empty set is missing.
2) As John Conway pointed out, we already have a perfectly good way to describe the set $\{1, 2, 3, \ldots \}$, namely the positive integers.  (JC was arguing why not to exclude zero from the natural numbers.) 
A: People really are free in mathematics to define things any which way they please.  For me, the phrase "natural numbers" is most natural when identifying them and thus defining them by equating them with the counting numbers, that is numbers that are used to count things (countable nouns).  Once you equate the phrase "natural numbers" with "counting numbers" it's most 'natural' (pun intended) to jettison zero, unless of course you consider the absence of something part of your counting system.  But in that case it's unsatisfying to me because, when considering countable nouns, you can only get a sense of absence if you first have a sense of presence.  And when you consider what is present, you count those items using {1,2,3,...}, NOT {0,1,2,...}.  So I would urge all mathematicians, not just some of them, to get a life and define N as = {1,2,3,...}.
