Georg Cantor
Are some infinities bigger than others?
That time, that place
Mathematicians had carefully avoided infinity. They used it to mean going on without end, but did not treat infinity itself as an object. Even Gauss objected to that usage.
Why this question
Cantor asked whether infinite collections can be compared in size.
His method is one a child knows. To compare without counting, pair them off. If every element finds a partner with none left over, the sizes are equal.
What was found
Pair the natural numbers with the even numbers: one to two, two to four, three to six. Nothing is left over. The evens look like half the naturals and are the same size. The part equals the whole.
Fractions, too, can be paired with the naturals.
The real numbers are different. His proof: suppose the reals have been paired with the naturals without remainder. Write the list down the page, then build a new number by changing the first digit of the first number, the second digit of the second, and so on. This number differs from every entry in at least one place, so it is not on the list.
Whatever list you bring, something is missing. There are infinities larger than others.
Infinity was not something you could ask the size of
The diagonal argument, producing a missing number from any list whatever
Kronecker and others attacked it fiercely. Cantor suffered from depression and was repeatedly hospitalized. Hilbert later said that no one shall expel us from the paradise Cantor created