Infinite dance - Lycée Arago (Perpignan) Colegiul National B.P. Hasdeu (Buzau, Roumanie)

You are organizing a big party with dancers. There are women, men and you want everyone to dance (no one sitting) in mixed pairs. Obviously this condition is possible if and only if there are exactly as many women as men.
Now imagine a party in a strange world in which you would have an infinity of men and women. Can you achieve the dance as you wish if, for example, there are as many men as natural numbers and as many women as integers (positive and non-positive)?
Imagine other situations with other infinite sets that you know: decimal numbers, rational numbers, real numbers...