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The mission of healthcare in the eyes of those who provide it, isn't "health at any cost".

For the people on the other side, "health at any cost" is pretty much the goal, usually limited by the "cost" side of things, especially in the parts of the world where they haven't yet figured out the whole "healthcare for the public" thing.

[fwip]

Cost here doesn't just include financial cost, but also time. As an extreme example, you could surely catch diseases earlier by visiting a doctor for an hour or two every day - getting tests for all sorts of things you might have conceivably developed. But that would make your life worse, and so most people wouldn't do that even if it was free.