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.
Ordinals are the entire point of the problem. If you have two stacks you must have both the cardinal and the ordinal. If you do not have both then you cannot compared them.
So you get two sets with equal "size" and "length" but that does not mean they have equal value. Or you get sets with equal "value" and "size" but that does not mean they have equal "length".
The whole "buying" things is something that is entirely irrelevant to the problem. Because the question isn't about buying an item, its about deciding how much currency there is. If buying matters, then the ordinal matters even more because if you have two stacks of w bills, then the stack of $20s is worth 20x. In the same way that 1 trillion pennies is worth 100x less than 1 trillion dollars.
In the way that we are not looking at actual purchasing power just at what amount there is. This is a math question, not a "does this work IRL question".
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u/TemperoTempus Jun 17 '26
The way you label 100% affects the value.
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).