They’re both countibly infinite thus the same, no?

To establish whether one set is of a larger cardinality, we try to establish a one-to-one correspondence between the members of the set.

For example, I have a very large dinner party and I don’t want to count up all the forks and spoons that I’ll need for the guests. So, instead of counting, everytime I place a fork on the table I also place a spoon. If I can match the two, they must be an equal number (whatever that number is).

So let’s start with one $1 bill. We’ll match it with one $100 bill. Let’s add a second $1 bill and match it with another $100 bill. Ad infinitum. For each $1 bill there is a corresponding $100 bill. So there is the same number of bills (the two infinite sets have the same cardinality).

You likely can see the point I’m making now; there are just as many $1 bills as there are $100 bills, but each $100 bill is worth more.

I think you’re misunderstanding the math a bit here. Let me give an example.

If you took a list of all the natural numbers, and a list off all multiples of 100, then you’ll find they have a 1 to 1 correspondence.
Now you might think “Ok, that means if we add up all the multiples of 100, we’ll have a bigger infinity than if we add up all the natural numbers. See, because when we add 1 for natural numbers, we add 100 in the list of multiples of 100. The same goes for 2 and 200, 3 and 300, and so on.”
But then you’ll notice a problem. The list of natural numbers already contains every multiple of 100 within it. Therefore, the list of natural numbers should be bigger because you’re adding more numbers. So now paradoxically, both sets seem like they should be bigger than the other.

The only resolution to this paradox is that both sets are exactly equal. I’m not smart enough to give a full mathematical proof of that, but hopefully that at least clears it up a bit.

Adding up 100 dollar bills infinitely and adding up 1 dollar bills infinitely is functionally exactly the same as adding up the natural numbers and all the multiples of 100.

The only way to have a larger infinity that I know of us to be uncountably infinite, because it is impossible to have a 1 to 1 correspondence of a countably infinite set, and an uncountably infinite set.

Infinity is a concept that can’t be reached so it can’t be counted up fully. Its not a hard number so you can’t get a full value from it since there is always another number to reach. Therefore you only peak at ∞ in any individual moment. You can never actually count it.

If you’re responding to the part about countable infinity and uncountable infinity, it’s a bit of a misnomer, but it is the proper term.

Countably infinite is when you can pick any number in the set and know what comes next.

Uncountably infinite is when it’s physically impossible to do that, such as with a set of all irrational numbers. You can pick any number you want, but it’s impossible to count what came before or after it because you could just make the decimal even more precise, infinitely.

The bizarre thing about this property is that even if you paired every number in a countably infinite set (such as a set of all irrational numbers) with a countably infinite set (such as a set of all natural numbers) then no matter how you paired them, you would always find a number from the uncountably infinite set you forgot. Infinitely many in fact.

It’s often demonstrated by drawing up a chart of all rational numbers, and pairing each with an irrational number. Even if you did it perfectly, you could change the first digit of the irrational number paired with one, change the second digit of the irrational number paired with two, and so on. Once you were done, you’d put all the new digits together in order, and now you have a new number that appears nowhere on your infinite list.
It’ll be at least one digit off from every single number you have, because you just went through and changed those digits.

Because of that property, uncountably infinite sets are often said to be larger than countably infinite sets. I suppose depending on your definition that’s true, but I think of it as just a different type of infinity.

Sure if you’re talking about a concept like money, but we’re talking about dollar bills and 100 dollar bills, physical objects. And if you’re talking about physical objects you have to consider material reality, if you’re choosing one or the other the 100 dollar bills are more convinient. Therefore they have more utility, which makes them have a higher value.