--- > The diagonal argument essentially starts with the question: What would happen if a bijection existed between the natural numbers and these real numbers? If such a function did exist, the two sets would have the same size, and you could use the function to match up each real number between zero and 1 with a natural number. You could imagine an ordered list of the matchings, like this. >  > The genius of the diagonal argument is that you can use this list to construct a real number that can’t be on the list. Start building a real number digit by digit in the following way: Make the first digit after the decimal point something different from a1, make the second digit something different from b2, make the third digit something different from c3, and so on. >  > This real number gets defined by its relationship with the diagonal of the list. Is it on the list? It can’t be the first number on the list, as it has a different first digit. Nor can it be the second number on the list, as it has a different second digit. In fact, it can’t be the *n*th number on this list, because it has a different *n*th digit. And this is true for all *n*, so this new number, which is between zero and 1, can’t be on the list. > But all the real numbers between zero and 1 were supposed to be on the list! This contradiction arises from the assumption that there exists a bijection between the natural numbers and the reals between zero and 1, and so no such bijection can exist. This means these infinite sets have different sizes. - [View Highlight](https://read.readwise.io/read/01haxwekrkfh9vyqa12kvppd7w) --- > Is ℵ1 the cardinality of the real numbers? In other words, are there any other infinities between the natural numbers and the real numbers? Cantor thought the answer was no — an assertion that came to be known as the [continuum hypothesis](https://www.quantamagazine.org/tag/continuum-hypothesis) — but he wasn’t able to prove it. - [View Highlight](https://read.readwise.io/read/01haxwgv4wjpx8yafrr89wn4xy) --- > In 1940 the famous logician [Kurt Gödel proved](https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/) that, under the commonly accepted rules of set theory, it’s impossible to prove that an infinity exists between that of the natural numbers and that of the reals. That might seem like a big step toward proving that the continuum hypothesis is true, but two decades later the mathematician Paul Cohen [proved](https://www.quantamagazine.org/to-settle-infinity-question-a-new-law-of-mathematics-20131126/) that it’s impossible to prove that such an infinity doesn’t exist! It turns out the continuum hypothesis can’t be proved one way or the other. - [View Highlight](https://read.readwise.io/read/01haxwhvfsk8taxq7c2cg5pbdr) ---