Ah, infinity. The ultimate childhood one-upper. The ace in the hole for when you really need to out do someone… times infinity. We all know what the word represents, but can it be practically applied? Let’s take a look at a few examples.

Hilbert‘s paradox of the Grand Hotel is the primary example of the counter intuitive nature of infinite applications. The initial concept involves a hotel with an infinite number of rooms, each of which is occupied. Then a new guest shows up and asks for a room. The hotel manager solves the problem by moving the guest from room 1 to room 2 and moves the guest from room 2 to room 3 and so on. Then the guest is able to check into room number 1 as it is now vacant and all the guests magically have a room again. I personally do not like this analogy as it involves adding a person to an infinite number of people. So I have modified it and will present you with the paradox below.

The Paradox of the Grand Lottery

Suppose we have an infinite number of people, each with a lottery ticket containing a number between 0 and 9. We then announce the winning number, 5 in our case. Each person has a 1 in 10 chance of winning so for the sake of simplicity let’s just assume that 1 out of every 10 people one. If we have an infinite number of people and each person has a single ticket, then we can conclude that we have an infinite number of tickets as well. Even though only 1 out of every 10 tickets are winners, we still have an infinite number of winning tickets. If this is true, then obviously even though not everyone has a winning ticket, it is still somehow possible for us to shift the winning tickets around so that everyone will end up with a winning ticket correct? This is the paradox, only 1 out of 10 people have a winning ticket but somehow there are still enough winning tickets for everyone. This is an absurdity of course, there will be infinitely more losers than winners even though there is an infinite number of each. Essentially, there are 10 times as many losers and still there are the same number of winners. It does not seem like this can be practically applied. In my opinion, if a theoretical concept can never be practically applied then it is an invalid concept, an illusion of oversimplification.

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