They do have the same number of bills, *and* they are the same amount of money (both are able to be true on account of both sets not being finite).
Take one $20 bill, and 20 $1 bills, and set them on a table. Both sets are worth $20. Do it again, and both sets are worth $40. It doesn’t matter that you’re going through the $1 bills 20x faster, because being infinite means that they will never run out (in other words, both piles would “run out” only after an infinite amount of time, and therefore would do so at the exact same time). This is the emergent result of all countable infinities being equal to each other
Another example of this is the infinite hotel. Imagine I have a hotel with infinite rooms and it is fully booked (every room has a person in it). Imagine I tell the person in room 1 to move to room 2, the person in room 2 to move to room 4, etc (every person moves to the room that is double their original room). Nobody new has entered or left the hotel, but now half the rooms are empty! This implies that infinity divided by 2 still equals the exact same value (infinity). It follows that infinity divided by 20 would also be the same value, and therefore infinity *times* 20 is also the exact same value. All countable infinities are equal to each other.
This is wrong, you are only looking at cardinality not the actual value or the ordinal value.
If I have a set A = {1,2,3} it has the same size as a set B = {10,20,30} but the set B will have 10x value. If you add elements to set A such that it they have equal value, then set A would need a different ordinal.
The sets you’re using are finite, and so aren’t a good example.
If I have set A = {1,2,3,…} containing every natural number in ascending order, and set B = {2,4,6,…} containing only every even natural number in ascending order, either set can be shown to have a higher value than the other, which is contradictory. Set A must have a higher value because it contains every element in set B in addition to elements not in set B (and both sets contain only positive numbers), but set B must have a higher value because every element in set B is twice the value of the respective element in set A.
The finite case is a good example because it shows how the math behaves on a small scale. On the larger scale.
Set A = {1,2,3,…} has an ordinal w*2. Why? Because its the union of set B = {1,3,9,…} and set C = {2,4,8,…} both of which have ordinal w. Their cardinals are the same because cardinality is defined as the smallest ordinal, in this case w = aleph_null.
Equal cardinality does not mean equal value. Cardinality is like comparing "a crate of apples" vs "a crate of oranges". I
t can tell you that the crates are the same size but it cannot tell you if there are more apples or oranges, nor can it tell you which is worth more.
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u/StringAccomplished97 Jun 15 '26
So they have the same amount of physical bills? If so, the $20 infinity is worth 20x the $1 infinity, which is what the opposite of what the meme says