ELI5: How are some infinities bigger than others?

The Cantor Diagonal proof is a way of determining countablility, yes. However, what the proof is trying to prove is that there is a place to start counting for a particular kind of set, because you can prove the identity of S1 and any particular Sn for that set, S. However, defining that mapping function is not always a walk in the park.

Also here's the function for the integers in JavaScript in case you're interested. I'm a professional programmer and even this one took me a good 5 minutes or so.

const mapNaturalToInteger = (natural) => {

if (natural === 0) { return 0; } else { const absoluteValue = Math.ceil(natural / 2); return (natural % 2 === 1) ? absoluteValue : 0 - absoluteValue; } }

console.log([0,1,2,3,4,5,6,7,8,9].map(mapNaturalToInteger)); // prints> [0, 1, -1, 2, -2, 3, -3, 4, -4, 5]

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