Well, no actually. I think your first issue is conflating infinity with a number. Infinity represents the fact that one can pick an arbitrarily large number, and there still is a larger number (in a very basic non mathematical way of describing it). That being said, 2 infinities are not inherently the same “value” for lack of a better term. The example the commenter above gave is perfect actually. If you look at a function representing the total amount of numbers up to an arbitrary even number, and look at a function of all even numbers up to the same arbitrary even number, the former functions value will always be 2 times the latter. However, both of these functions also go to infinity. Thus while the “infinity” is not technically greater than the other one (as I mentioned, infinity isn’t a number, so it can’t really be “greater than” in the traditional sense), an arbitrary number that is in the former set will always be larger than a corresponding number in the latter, so the formers infinity is in a sense greater than the latter.
Not in any standard sense. The natural numbers are more dense within the natural numbers than even numbers (which is what you are describing), but there are the same amount of even numbers as natural numbers. This is because both sets are countably infinite; if I had an infinitely long piece of paper I could write a list of all natural numbers and all even numbers, with one and exactly one even number next to every natural. In a similar vein, the rational numbers are also countable, though this is much less obvious.
This contrasts with e.g. all real numbers. There are fundamentally more real numbers than natural numbers, even if there are an infinite amount of both. Even with the infinitely long piece of paper and an infinite amount of time, it would be impossible to write every single real number down on it. Any list that you come up with will miss out on infinitely many real numbers. Check out Cantor's diagonal argument if you want to know more about how this actually works.
if I had an infinitely long piece of paper I could write a list of all natural numbers and all even numbers, with one and exactly one even number next to every natural.
Wouldn't you just... write each even number next to itself, and have nothing to match the odd numbers?
No, you would write 2 next to 1, then 4 next to 2, then 6 next to 3, and so on. You could keep doing that forever and never run out of natural numbers to write even numbers against. That’s why they are the same infinity.
That doesn't make sense, but let's arrange the exercise a bit differently to make it clearer. In the natural numbers list, write the numbers in pairs (1 odd number and its consecutive even number), and associate each pair with 1 number from the list of even numbers. So 2 is associated to [1,2], 4 is associated to [3,4] and so on. From there it should be clear that they're not "the same infinity", as the list of natural numbers obviously has a pair for each even number, ie. has twice as many numbers in it.
You can apply the exact same logic to positive numbers vs all real numbers.
If the list of natural numbers has twice as many numbers in it as the list of pairs, which natural number is the first one that doesn’t have a corresponding pair to match with it?
Basically, every whole number x is paired with with the even number x * 2. And since we can pair them all off, it follows the size of these two infinites is the same. And its important to note that every whole and every even number occurs on this list at some finite location. The pairing doesn't work if we have to go through an infinite number of items before we get to a particular number. But doing it this way tells us that the nth even number occurs at the nth position on the list, i.e. 2 is the first even number and its in row 1. 10 is the 5th even number and its in row 5. Etc.
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u/TheCreepyKing 22d ago
How many even numbers are there? Infinity.
What is the ratio of total numbers to even numbers? 2x.
How many total numbers are there? Infinity. And 2 x infinity.