Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?
In: Mathematics
Some infinities seem bigger than others because, well, they are!
When we count sets of things, we create a one-to-one mapping between a that set of things and a set of objects whose size, or “cardinality”, is known. A one-to-one mapping is just that: each object in set A has exactly one corresponding object in set B, and vice-versa. A simple example is counting on our fingers: we know that we have ten fingers, so if we can match each object to one of our fingers, we know that we have ten objects in the set which we’re counting. In mathematics we attempt to find some mathematical function that creates this mapping for us.
The “smallest” infinity is the Natural Numbers: 1,2,3,4, … and so on. Consider the set of Real Numbers: things like -1, π, 0, 1,000,000, and so on. Can we find some mathematical function that gives us a one-to-one mapping between the Natural Numbers and the Real Numbers? Spoiler: no, we can’t, so those two infinite sets are not the same size.
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