> Things become worth more as they become scarecer, not less.

It's a bit more nuanced than that. Generally speaking, in society as we know it, there's a point at which something becomes so scarce that it's value begins to drop because there's simply no use for it and no reason to find a use for it.

The second part to this is that scarcity-based value is a product of our current society, and we're describing a society that diverges from that. This could go in many different ways, some that could well be a massive improvement to what we currently have, but for the sake of this argument, imagine everything goes very badly and people are starving to death. Would you trade a weeks worth of food for a diamond?