Does the set of natural numbers contain infinity? Let $N=\{1,2....\}$ be the set of naturals numbers. Is $\infty \in N$, as the list is never ending  ?  
 A: One property about the set $\Bbb N$ of natural numbers is that it does not have a largest element, so if you defined $\infty$ as some element which is larger than every natural number, it wouldn't be in $\Bbb N$ anyway because of this property.
Having said that, I recommend you look up the ordinal numbers and what it means for a set to be well-ordered (it means every subset has a least element).  We could add an element into the set $\{1,2,3, \dots \}$ which we define to be larger than every element in $\{1,2,3,\dots \}$.  Usually, instead of $\infty$, we call the element $\omega$.  But the set $\{1,2,3,\dots, \omega \}$ is a different set than $\Bbb N$ (it contains $\Bbb N$ as a proper subset).  $\omega$ is the largest element in this set, and you could think about $\omega$ as "infinity" in some sense.
But then we can add an element defined to be larger than every element in this set.  Let's call this element $\omega + 1$.  So $\{1, 2, 3, \dots, \omega, \omega + 1 \}$ is the set where $\omega$ is larger than every natural number, and $\omega + 1$ is larger than every natural number and larger than $\omega$.  So, it's not useful to call $\omega$ "$\infty$" because there is something larger than it.  (Although, Georg Cantor proved that, as far as set cardinalities/sizes go, there are infinitely many distinct infinites...).
A: No, if $\infty \in N$, then the list would have an end, namely $\infty$. Of course, you would still need to go infinitely far up from $1$ (or indeed any other number) to get to $\infty$.
A: Infinity is not well-defined as it has different meanings depending on the context so placing it in a set that is well-defined (the set of natural numbers $\mathbb{N}$ for instance) is clearly meaningless. 
