What is particularly interesting though is that while both infinite sets, the integers have an infinite size that’s smaller (in a precise sense) than that of the real numbers. More generally, if one (finite) set is larger than another, then we can always relabel the larger set so that the smaller one becomes a subset of it. Let’s now assume that this property continues to hold for infinite sets (or, if you like, we can use this very natural property as part of the foundation for the definition of the sizes of infinite sets). First, we observe that the integers are a subset of the real numbers, and hence cannot have size larger than the real numbers. For example, we can define so that it satisfies (for all real numbers x):when x > 0Similar rules could be used to define .

When you want to outsmart the world, you turn to the facts. And the facts are in the science.