Is it impossible to pick 7 (or some number) from a cardinally infinite set? (Aleph-0) It should be said, I have no background in calculus, but have read a few books on infinity and still don't understand it's relation to statistics.
From what I've read it sounds like the textbook answer is something along the lines of:
"The probability of picking 7 is indistinguishable from 0"  
Which seems to read as though it's impossible to pick 7. And of course 7 can be swapped out with any number in the infinite set.  Which leads me to the conclusion that the probability of picking any number from an infinite set is indistinguishable from 0.  This, though, seems to be able to be flipped on it's head by simply asking what the probability of picking a number contained by the set via randomly picking some positive integer.  
Can someone help me make sense of this?
 A: It's totally a question of what probability distribution you're putting on this set (which I'm guessing is either the integers or natural numbers in your example). It can't be uniform because $\aleph_0$ refers to countable infinities and a sum of countably-many of the same nonnegative number is either zero or infinite (it can never equal one).
But all you need to do is spread out the probability in another way. Some simple examples are the standard discrete distributions you encounter like the Poisson or geometric (as whuber has already pointed out).  If $X$ is our selection from $\mathbb{N}$ say, then in the Poisson case we take $P(X = k) = e^{- \lambda} \lambda^k / k!$ for some $\lambda > 0$, and in the geometric case (depending on how we decide to define things) we have $P(X = k) = (1 - p)^k p$ where $p \in (0, 1)$.  In both cases $k$ can be any natural number whatsoever, so we have no trouble selecting 7 or any other value.  The problem only comes in when we insist on trying to choose things uniformly at random.
A: When you start playing with infinities phrases like "indistinguishable from 0" are not the same as "equal to 0."  In this case the probability is being handled as a limit.  One could consider the probability of 7 being picked is $1/\aleph_0$, however, $\aleph_0$ is what is called a "limit cardinal," and infinitesimals generated from limit cardinals are not included in the set of real numbers, so we have to be creative.
Consider a series defined by $1/x$ as we vary x:
x    1/x
1    1
2    0.5
3    0.3333...
4    0.25
...
10   0.1
100  0.01
1000 0.001
...
ℵ_0  ?

So what's $1/\aleph_0$?  It can't be zero, because that doesn't satisfy the rules of arithmetic $0 * \aleph_0 = 0$, but basic reorganization of the algebra shows that $1/x * x = 1$ for all x ($x \neq 0$).
The concept of a limit is used to rigorously explore this intuitive understanding that there is some limiting thing out there for $1/x$ that equals 0.  In very intuitive and imprecise wording, this limit is the number "approached" by a series, but typically never actually achieves it.
Thus, when given an infinite number line and asked to pick a random value from it, the probability of any given number being picked is very very small.  In fact, so small that the real numbers can't even describe how small it is.  To capture this concept of smallness, we often look at the limit of the probability of a number being picked as we grow the set towards a cardinality of $\aleph_0$.  This limit is zero.  This is not to say the probability of picking any given number is zero, just that any process used to describe the meaning of picking randomly from an infinite set that starts from smaller sets and builds larger ones from there (typically using induction) will describe a series of probabilities that get closer and closer to 0 as the sets get larger.
The more formal definition of this limit could be phrased with some of the calculus you mention, using the epsilon-delta definition of a limit.  By this definition, if you pick any arbitrarily small "epsilon," and try to determine which sized sets result in a probability of picking 7 below this epsilon, I can identify some set size for which all larger sets must yield probabilities smaller than this epsilon bound.
Of course, you probably don't need the formal definition much.  Sometimes intuition is enough, but its good to know there's a formal definition coming later down the line for you!
A: Think about it this way: Randomly picking a number means that you will have a number (some number) after picking.  If it were true that it's impossible to pick the number N from the infinite set, then any number you would have picked would lead to a logical contradiction.
So the real question is whether you can choose integers at random with an even distribution from an infinite set.  I'd say the answer is no, not in 'reality'.  In order to do so you would need a process by which you have an equal probability of selecting any number from an infinite set.  This would require (at the very least) an infinite amount of time and unfortunately, that is not available to us.
For example, let's say you roll a 10 sided die (0-9) some number of times to pick a random number.  How many times you roll it will set the upper bound of what number you might get.  If you roll it three times, your number will come from the set 0-999.  You'll be rolling the die forever since there is no limit to the number of digits in the set.
