What is Ross-Littlewood paradox?

The Ross-Littlewood paradox, also known as the "Littlewood's paradox," is a thought experiment that highlights some intriguing features of infinite sets and their properties in set theory and mathematical logic. It revolves around the concepts of countable and uncountable sets. The paradox is presented as follows: 1. **Setup**: Imagine a sequential process in which balls are placed into a box. At each second, a certain number of balls is added to the box: - At the first second, one ball is added. - At the second second, two balls are added. - At the third second, three balls are added. - And so forth. After infinite time has passed, all the balls added form an infinite collection. 2. **Removal**: Now, consider that the process of removing balls from the box happens simultaneously. Each ball can be removed according to its index: - In the first second, the first ball is removed. - In the second second, the second ball is removed. - In the third second, the third ball is removed. - And so on. 3. **Result**: The paradox arises when we consider what happens after infinitely many balls have been added and removed. It leads to the conclusion that if the process is carried out carefully, it may appear that you could still have infinitely many balls remaining in the box after all attempts to remove them have been made. This presents a conflict or paradox regarding the nature of infinity and counting. The Ross-Littlewood paradox serves to illustrate the counterintuitive properties of infinite sets, particularly in how they can yield seemingly contradictory or paradoxical outcomes. It highlights the challenges and nuances in understanding the concept of infinity in mathematics.