TITLE: is "N -> N where g(n) = all integers > n" a function?
QUESTION [0 upvotes]: I know this is probably not the best question, but I'm sitting on the fence about my answer because I'm not 100% sure of "Natural Numbers" definition. Does it include negatives or is it strictly positive? I'm just a tad confused for when I Google it the examples always include Z. ex. http://mathworld.wolfram.com/N.html 
This is one side of the fence...(note: I'm not including 0 as a natural number) 
"Not a function because g(1) has more than one value." (since it could be -ve or +ve?) 
and this is the other..
"Is a function because g(1) is the lowest value since all others are greater" (If N only includes positive numbers)

REPLY [1 votes]: The problem is that

$\Bbb N\to\Bbb N$ where $g(n)=\text{all integers }>n$

doesn’t define anything. The notation $\Bbb N\to\Bbb N$ indicates that you’re talking about a function whose domain is the set of positive integers, and that returns outputs that are also positive integers. That means that for each positive integer $n$, $g(n)$ has to be a single specific positive integer. It can’t be a set of positive integers, because a set of positive integers isn’t an element of the codomain (‘target set’) $\Bbb N$. It can’t some indeterminate number greater than $n$, because it must be a specific value.
You could define a function $g:\Bbb N\to\Bbb N$ by setting $g(n)$ to be the smallest integer greater than $n$; that’s the function $g(n)=n+1$. You could also define a function $g:\Bbb N\to\wp(\Bbb N)$ from $\Bbb N$ to the power set of $\Bbb N$, i.e., the set of all subsets of $\Bbb N$, by setting
$$g(n)=\{k\in\Bbb N:k>n\}\;,$$
i.e., by letting $g(n)$ be the set of all positive integers greater than $n$. This is a fine function; it just isn’t a function from $\Bbb N$ to $\Bbb N$.
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