One example of such paradox can be illustrated as follows (this is not mentioned in this documentary):

Let's start with the digits 1 and 2. Let halve these two digits to get 1/2. And then we halve that, and so forth to ad nauseum, so we get the series,

1, 1/2, 1/4, 1/8, 1/16 ...1/∞

Mathematician will tell you that this series has infinite members or fractions, and yet we know that it contains within the finite boundaries of 1 and 2. Finite, and yet infinite. This is in fact the well known Zeno's Paradox that originated in the ancient Greece.

Other seemingly mind-boggling concept that does appear in this documentary is the sets of equations:

∞ + 1 = ∞

∞ + ∞ = ∞

∞ - 1 = ∞

∞ - ∞ = ???

We assume that the ∞ in the equation belongs to a positive digit number.

The first three equations seems to make sense 'intuitively', and at the same time seems contradictory because how can z + 1 = z ? This equations only 'makes sense' if z = ∞.

Similar 'logic' applies to the next 2 equations.

What about the last equation (∞ - ∞ = ???) ? According to the mathematician in the documentary, it could be 1, it could 3, it could be anything ! Nobody knows. Well, this is no good.

Here's what I think is wrong with the picture.

when you have the equation 1 + z = ?

It doesn't matter how big z is as long as it isn't ∞, I can give you a precise answer. If z = ∞, then I can' give you an answer. Why? Because ∞ isn't a number. While we think of it as an infinitely large number, but it is not! It's a concept, an idea. So it's nonsensical to apply an arithmetic operation to a concept.

The operation 1 + ∞ makes as much sense as 1 + democracy, or 1 + chaos.

'∞' isn't a number, so you can't apply mathematical operation to it.

Too simple? Or maybe the documentary had made a poopoo with that sets of equations, or that I've missed something.