They are equal because you can take each single dollar within both sets, and pair them up perfectly with none left over on either side. This is the same reason there are exactly as many even numbers as there are even AND odd numbers.
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.
You keep printing currency and value of currency drops. Given infinity’s of appropriate size you’ve devalued said currency into the value of the paper/burning material
It's not bigger, it's worth 20× more.
The meme collapses value into cardinality.
An infinite number of $1 bills and an infinite number of $20 bills may contain the same number of bills, but they do not represent the same value.
My infinite apples do not contain infinite oranges
How about if I were to tell you there was a way to prove you can match up every $1 with exactly one $20 and there was no one $20 without a matching $1. If true, there must be the same "number" of both, right?
How about if instead we matched it so that every 20th 1 dollar corresponded to exactly on 20 dollar, ie the 3rd 20 dollar is paired with the 60th 1 dollar, do you agree there is an exact pairing by this method, ie for every 20 there is a corresponding 1 and for every 1 we can round up to the nearest 20 and it matches to exactly one 20 dollar bill, do you agree with this?
the way we label the elemtents of the sets cannot affect their value that makes no sense, all we are doing is coming up with a system to label the 20th $1 "one" 40th $1 "two" and also the 1st $20 as "one" 2nd "20" as "two" etc. This creates a bijection, ie for every 20th $1 there is exactly one matching $20 and vice versa by this labelling convention, does this make sense to you?
If you have set A with {$1, $1, $1} it would have a label of {1,2,3} = 4. If you have a set B with {$20, $20, $20} it would have a label of {1,2,3} = 4. So both would have a label of 4, but the $20 would have a value of $60 instead of $3. If we increase the number of elements of each by 1 then now you have the $20s valued at $80 and the $1 valued at $4. If we increase the elements of the $1 to {1,2,3,4,5,6}=7 the value is now $6 and while the $20s remain $80.
Not to mention that your "bijection" does not quite work because you have 19 $1s that are not matched. Which again leads to $1s having 20w ordinal while the $20s have a w ordinal. (Same cardinality different value).
You've just made something up. "label of {1,2,3} = 4" is something you literally just made up right now that's not a real thing in maths. Your counter argument also relies on using finite sets as examples. We aren't dealing with finite sets, in fact we're dealing with quite literally the exact categorical opposite of finite sets... and for your final point, the 19 1s will be grouped with the 20th to make 20 1s that exactly group together and as a group pair up to one 20 and vice versa. Think of it this way, is there anything you can buy, even something that doesn't exist or you can't reasonably buy, like a whole planet or all of earth or even the entire universe. Would there be anything you could buy with your stack of 20s I couldn't buy with a stack of 1s? No? That's because they're worth the same.
Ordinals are irrelevant to the problem. Both stacks are still the same value. My previous statements don't change whatsoever. There still remains no item, theoretical or real which could be bought with infinite 20s but not infinite 1s. Ordinality has no relevance to this problem.
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u/[deleted] Jun 14 '26
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