Actually, studies have shown that in Europe (possibly the US too) the educated classes are reproducing less and less. Since it is usually the educated class that generate more than a middle class income, what we would probably see is a new societal rift rather than overpopulation. Since this health care would presumably be costly then the poor would continue to die while the upper class continue to live. This could be potentially volatile, and as long as the healthcare is not accessible to exclusively one country then it would probably lead to massive rioting on an international level. Heck, it could be the thing that triggers a global communist regime.

However, if countries manage to fix the health care system and make this accessible to all working citizens then rioting would be less of an issue, while overpopulation could become a potential one. China has been addressing the overpopulation problem for some time now, so other countries would simply have to take their lead. This could cause problems for the people of the Catholic and Mormon faiths, who tend to encourage large families.

Also, this gives an interesting twist to the anti-biblical argument that biblical characters could not have lived to be as old as the bible reports (I believe Noah was reported to have lived over 200 years.)

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"A short time ago you agreed that knowledge and opinion are not the same............
"How can any intelligent man say that a fallible thing is the same as an infallible one?"