Nothing = Infinity

This is a question from an elementary math book:

[b] u = u + 1.
(i) Find the value of u
(ii) What is the difference between nothing and zero?[/b]

If you try and solve u = u + 1 you'll get 0 = 1 (subtracting a from both sides)

0 = 1 is a contradiction. So u is nothing. u is NOT zero. u is nothing.

Why?

Take the equation below:

e + 1 = 1

Solving the equation for e gives us e = 0. The same cannot be said of u = u + 1 our first problem.

So given the above equations ( u = u + 1 AND e + 1 = 1) we have the following:

1) u is NOTHING. u is NOT zero
2) e = zero

What's the difference between NOTHING and zero?

My "explanation" is in terms of solution sets.

The solution set for u = u + 1 is the empty set { } with no members
The solution set for e + 1 = 1 is {0} with ONE member viz. zero.

There's another mathematical entity that can be used on the equation u = u + 1 and that is INFINITY.

INFINITY + 1 = INFINITY

So we have:

a) u is NOTHING
b) u is INFINITY

Therefore,

NOTHING = INFINITY

Where did I make a mistake?

Thank you.