By: Vinayak Prathikanti, Varun Popli, & Alan Xiao

For the second month, we had the Infinity POM, and it definitely made us think a little. The problem said to consider all of the infinite numbers between 0 and 1, then all of the numbers on the number line. Which one of these is the larger infinity? Now before we say anything else, we want to tell you that this is an extremely open ended problem, and that it is very hard to argue one side. While this is true, we want to try to cover all of the ways that you could think about it, then express our own opinion at the end. Just to clarify, when we say “which infinity is bigger” we mean “which infinity contains more numbers”. Now, we’re big fans of solving math

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Then, think of all of the integers from 0 to positive and negative infinity, so no 2.125 or 55.2329879. This is a different number line/situation than the problem, but we think that it will give you a better understanding of infinity and also support our opinion better. I’m going to show you how we can use the 1-1 correspondence to explain which infinity is bigger in this situation, if any. But how can we assign numbers to numbers if we can’t start at a number for both infinities? For example, we can assign the number 1 to 0.1, but that would neglect the infinite amount of numbers smaller than 0.1, and on top of that you could only stop at the integer 10 after you assign it to 1.0, neglecting the infinite integers larger than 10, and we can’t have that! So here’s what we can do: Assign 1 to 1, add 1 to the integer side and divide 1 by the number on the integer side. It sounds confusing at first, but it’s very simple in numbers: The next step after 1 to 1 would be 2 to ½, then 3 to ⅓ then 4 to ¼, and so on! By doing this, we can prove that it is possible to assign every integer to a fraction. Now if we weren’t “thinkers” we would stop there, say “Alright, since we can assign every integer to a fraction, these 2 infinities are equal”, and be done. But since we are thinkers, we say, “But wait, there’s

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First off, it is impossible to assign a variable to a group of number that is infinite and keep going on forever. With that in mind, it would be impossible to count how big the whole line is because we could not count those variables because they represent and incorrect amount of number between 0 and 1, so our first solution is false. For our second solution, we can render that false as well because that one to one correspondence cannot keep going on forever, as we cannot count that far. Also, as we stated in that section, we know that we cannot perform that one to one correspondence on the infinite fractions in between 0 and 1. With all these in mind, we came to the conclusion that both of the infinites are equal.

Conclusion In the end, we can make a case for all possible infinities that are greater, but personally we believe that they are equivalent. There are always so many other ways to prove any of these cases, but for us, it all comes down to logic, like we said earlier. Infinity is a concept, an idea, not a countable number, so we can surmise that they are