The root word of wealth is well. Wealth therefore means wellness. In societies based on money, money becomes equated with wellness.
...I don't see how this follows...
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Who wouldn't want more wellness? Particularly if you can keep it and hold on to it.
And I certainly don't see how this follows.
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When the amount of money is finite, and people want as much as they can get, then there simply isn't enough to go around. Every gain on your part is a loss on someone else's part.
Wouldn't the very first statement be true in any given system in which anyone is capable of owning anything at all?
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No. Money (and possessions) need not equate to wellness.