The problem seems to contradict itself and that's why it puzzles me. Here's the full text:
Let there be an infinitely sided die where each face has an equal probability of appearing upon casting the die. Each side is marked $\{1,2,3,\ldots\}$. What is the probability that upon casting this die a multiple of 5 appears?
How can this die form a uniformly distributed set?
Let p be the probability of rolling each number. Then $\sum p = 1$, but there are infinitely many $p$'s, so $p=\lim_{n\to\infty}\frac{1}{n}$, which leads to $p = 0$, which, in\ turn, contradicts $\sum p=1$.
I know I am missing something, but what is it?
It is impossible for a random variable $X$ that takes a countably infinite number of values $\{1,2,...\}$ to be uniformly distributed. To see this, imagine $$ \exists \ p \in [0,1] \ \ \forall i \in \{1,2,...\}: P(X = i) = p.$$ Then we immediately get the contradiction: $$ 1 = \sum_{i=1}^{\infty} P(X=i) = \sum_{i=1}^{\infty} p \in \{0, \infty\}. $$ So your problem is indeed ill-posed since the die your describe cannot possibly exist.