> When I said they are not the same size, I did not mean
to invoke the idea of set cardinality.
Set size and cardinality are one in the same. You would have to change the axioms of set theory in order to define size any other way.
> if set A contains all the elements of B and set A
contains other elements also, then set A is bigger than
set B
Not if you take "bigger" to mean "of greater size." This statement is only true if "is bigger than" means "includes" (= "is a superset of"). Set inclusion and set size are distinct concepts. We can say that if set A contains all the elements of set B, then set A is at least as numerous as set B. But we cannot say that if set A additionally contains elements that set B does not, then set A is more numerous than set B.
> In my way of thinking, the cardinality of a set of
dollar bills and the quantity of dollars are not the
same thing.
They are certainly distinct concepts, but for a countably infinite set of bills with bounded face values, the two quantities are equal (even if they are computed differently).
> if you start with an infinite quantity of dollars
and lose $20, then you lost $20
Let us distinguish money, the sum of the nominal value, from currency, the specific expression of that value. If you lose $20, then your currency has changed (let's say you lost a specific $20 bill), but your money has not (you can keep losing $20s without any meaningful consequence).
> Whether the set of dollars before and after gambling
have the same cardinality is quite beside the point: $20
never equals $0, so you were $20 richer before you
gambled and $20 less rich after you gambled.
How do you define "rich"? Do you define "rich" to be "in possession of a large
quantity of money" or do you define "rich" to be "in possession of a
specific set of dollar bills"? They are not the same. I don't care if my $20 bill has serial number X or serial number Y, it's still worth $20 to me.
Likewise, although you can measure the value of the difference between the set of money you had before and the set of money you have after, as long as that difference is finite, it has no bearing on your infinite total value. In fact, depending on how you do it, you can lose an infinite amount of money and still have an infinite amount left (the difference between {1, 2, 3, ...} and {2, 4, 6, ...} is {1, 3, 5, ...} and all three of these sets are equinumerous).
Infinity is not an intuitive concept.