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The set of integers and natural numbers have symbols for them:

  • $\mathbb{Z}$ = integers = {$\ldots, -2, -1, 0, 1, 2, \ldots$}

  • $\mathbb{N}$ = natural numbers ($\mathbb{Z^+}$) = {$1, 2, 3, \ldots$}

Even though there appears to be some confusion as to exactly What are the "whole numbers"?, my question is what is the symbol to represent the set $0, 1, 2, \ldots $. I have not seen $\mathbb{W}$ used so wondering if there is another symbol for this set, or if this set does not have an official symbol associated with it.

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    $\begingroup$ There are no official symbols (literally, in that there is no office that decides these things). I use $\mathbb{Z}_{\ge 0}$. $\endgroup$ – Qiaochu Yuan May 29 '12 at 5:25
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    $\begingroup$ For someone in logic, often $\mathbb{N}$, or $\omega$. $\endgroup$ – André Nicolas May 29 '12 at 5:25
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    $\begingroup$ My personal preference is to call $\{1,2,3,\dots\} = \mathbb{Z}^+$ (or $\mathbb{Z}_+$ where there might be confusion with the additive group) and use $\mathbb{N}$ for $\{0,1,2,3,\dots\}$, but as Qiaochu said, there's no real convention about this. If you prefer your form of $\mathbb{N}$, you could use $\mathbb{N}\cup\{0\}$ like one of my profs. $\endgroup$ – Eric Stucky May 29 '12 at 5:27
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    $\begingroup$ $\sqrt{\mathbb{Z}^2}$?, $|\mathbb{Z}|$? Actually I use $\mathbb{N}$ for $\left\{0,1,2,\ldots\right\}$, so to each their own. $\endgroup$ – alex.jordan May 29 '12 at 5:47
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    $\begingroup$ Also, sometimes $\mathbb{N}_0$ is used for $\{0,1,2,\dotsc\}$. $\endgroup$ – Miha Habič May 29 '12 at 5:56
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To summarize what has been said in the comments, there are no "official" symbols. Use whichever notation you feel most comfortable with, as long as it makes sense and can be easily understood by the general audience.

Some examples include:

$\mathbb{Z}_{\ge 0},\mathbb{Z}^{+}\cup\{0\},\mathbb{N}\cup\{0\},\mathbb{N}_0$

Also note that because of different conventions, what you refer to as "whole numbers" may or may not include zero. From Wikipedia:

There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}.

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  • $\begingroup$ Yeah I am aware of the confusion about whole numbers, so had specifically listed the set I wanted to name. But, also based on this I am going to use $\mathbb{N}_0$, but good to have a list of possible other symbols as a reference. $\endgroup$ – Peter Grill May 31 '12 at 19:19
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Except $\mathbb{N}\cup\{0\}$ we use $\mathbb{I}$ symbol too for $\{0,1,2,3,...\}$ and we call it the set of calculating numbers that it has only zero more than natural numbers. Also in some books it has been denoted by $\mathbb{Z}^{\geq0}$

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