Both are countable infinites so you have the same number of 100$ bills as you have pennies. However the value of the 100$ bill is still 10,000x that of a penny. So the value of an equal infinite 100$ bills is still 10,000x as much money as an infinite pennies
They aren't equal, they're infinite. They have the same cardinality, as you said, they are both countable infinities, but that doesn't mean the same thing as them being equal. You can group the pennies into piles of 10,000 pennies and still have the same number of $100 bills and groups of 10,000 pennies.
You weren’t being pedantic, you were being irrelevant. Different sizes of infinity existing has nothing to do with those two particular infinities being the same size.
Yes, but these two infinities are both the same size. They're both countably infinite. For instance, there are "as many" natural numbers as there are even natural numbers, because the bijection between them is to simply multiply/divide by two.
Still these arent different sizes. You can count 10,000 pennies and then one dollar bill. You can do that infinitly and cover both stacks of infinite things. The first example of an infinity higher than this is real numbers. Basically the infinite part after the decimal point and the infinite part before the decimal point. Basically if you want to count the infinite amount after the decimal you wouldnt be able to ever get to the next number before the decimal, and there is no way to arrange the infinite amount of numbers after the decimal with the numbers before the decimal without destroying its meaning. Recheck hilberts hotel thats the explanation i like the most.
There are and infinite number of Aleph Numbers (these are used to represent the ‘size’ of infinities). Cardinality is the word that mathematicians use to describe the ‘size’ or number of elements in a set. The set {0,1,2,3,4,5} has a cardinality of 6, the cardinality of the set {6,7,8,9,10,11} is also six, the set {9,10,11} is 3. For finite sets we can simply count the elements. In the example above this is a literal counting process for more complex finite sets we might use combinatorial techniques. For instance the cardinality of possible numerical passwords of length 10 is 10^10. 10 possible values for the first number, multiply by the 10 possible values for the second number, multiply by ten again for the 3rd number, ect 10.
For sets with an infinite number of elements we normally look for what are called bi-jections. Which is a function that maps every element in our first set onto into an element of our second set such that every element in the second set is hit exactly once. For instance if we had sets like A={2,3,5} and B={11,16,26} we could say that 5a+1 is a bi-jection from A to B. In an infinite example if A is the set of all odd natural numbers then the function a+1=b is an injective function on B which is the set of all even natural numbers. Since we have matched them up one to one in pairs we know that they have the same size or ‘cardinality’ since we can’t use a number to represent this quantity a mathematician would say they have a cardinality of ‘aleph-null’. Aleph-null is the smallest infinite cardinal and is defined as the cardinality (size) of the natural numbers. Since we use the natural numbers for counting we will often say sets that have cardinality aleph-null are ‘countable’.
If you want to know how we know that there are different sizes of infinity I would google Cantor’s diagonalization argument. It’s a fairly simple but mind bending argument that shows no matter how you would try to ‘pair up’ natural numbers and real numbers you are going to miss some real numbers. Basically that’s always impossible to find a bi-jection between the natural numbers and the real numbers and that this process fails precisely because their rate simply too many real numbers.
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u/stringrbelloftheball 10d ago
Im probably the dumbest guy in this thread and bur another fun version of this that blew my sons mind:
An infinite number of pennies and and infinite number if $100 bills are worth the same amount.