Is getting a random integer even possible? On googling I got


  Random numbers are numbers that occur in a sequence such that two conditions are met: (1) the values are uniformly distributed over a defined interval or set, and (2) it is impossible to predict future values based on past or present ones. Random numbers are important in statistical analysis and probability theory.


What if I want to pick a random integer from the number line ?
The number line consists of infinitely many integers.
My favorable event is to get a number (random) and the number of events possible is the complete number line i.e. infinity.
I won't be considering any limits here as our number line doesn't tend to infinity but does contain infinitely many elements.

Now the probability of getting a random number

P(E) = 1 / ∞ = 0


So I won't be able to get a random number, ever.
This implies, never would I ever be able to get a random number. 
That's absurd as many of our science topics do depends on things being random.
Most typically you choose a random natural from a finite range.  If you toss a fair coin, you get a random number that is either $0$ or $1$.  If you call a random number generator on your computer, you might get a number in the range $[0,2^{32}-1]$.  There is no problem having equal probability in these cases.  If you want to choose a random natural, you can't have the probability of all numbers be equal, but you can use a non-uniform distribution that sums to $1$.  If you don't think $0$ is a natural, you can have $p(n)=2^{-n}$, for example.
Progress through the integers beginning with the first element in a well-founded wellorder and combine that with a probability $p_x=f(x)$ of stopping on any given element, given that you land on it, where $f(x)$ satisfies $\forall x:0&lt;f(x)&lt;1$.

Then by this means you can not only do so, but given an appropriate function, infinitely many integer outcomes (including possibly every integer) can have non zero probability.

An appropriate function for the natural numbers $\Bbb N^+$ is $f(x)=1/2^x$