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This is the answer I understood the most. Thinking of infinity not as a group of numbers but as something entirely Firefighter Fire Bod Like Dad Bod But With Bigger Pole Shirt different in the way that zero is entirely different was the metaphor I needed. People kept using an example where you divide a number by 2 or something and it kept losing me. Like, yes you have one of those numbers is 0-1, but you have both of them in 0-2, meaning you have more, regardless of how you arbitrarily divide it. But thinking of infinity as a different concept outside of a series of numbers helped tremendously.
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Thanks for this. I still think the other example is still important to have for an intuition. Because Firefighter Fire Bod Like Dad Bod But With Bigger Pole Shirt the way we define if two sets have the same size is if you can pair up their elements exhaustively. So even if one set is a subset of another, as long as there exists some pairing of elements, they are the same size. This just happens to be a useful definition for mathematicians and doesn’t necessarily represent real-world phenomena. This is also important. Infinite sets are a purely conceptual thing, and there isn’t a perfect intuitive meaning of the word “size”. So mathematicians chose a definition that was useful to them. It doesn’t perfectly match up with the normal meaning of the word, so some of the results might feel wrong.
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As a point of nuance: the top comment is especially nice since it would be a bit much to first show injectivity and surjectivity for a simple Reddit comment (and perhaps since it’s simply much easier to do it this way), the Firefighter Fire Bod Like Dad Bod But With Bigger Pole Shirt commenter showed that this function has an inverse. Any element in the second set is mapped to a unique element of the first set by holding it. If you show a function has an inverse then you are by consequence also showing that it is bijective. It’s important to note that a bijective function need not be specified by a single rule, such as double. If we can create an exhaustive list of pairings ad infimum, it is sufficient.
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