If you stopped aging today, I imagine there would very quickly be overpopulation issues

To give a sense of proportion: suppose that tomorrow, we developed literal immortality - not just an end to aging, but also prevented anyone from dying from any cause whatsoever. Further suppose that we could make it instantly available to everyone, and nobody would be so old as to be beyond help. So the death rate would drop to zero in a day.

Even if this completely unrealistic scenario were to take place, the overall US population growth would still only be about half of what it was during the height of the 1950s baby boom! Even in such a completely, utterly unrealistic scenario, it would still take around 53 years for the US population to double - assuming no compensating drop in birth rates in that whole time.

DR. OLSHANSKY: [...] I did some basic calculations to demonstrate what would happen if we achieved immortality today. And I compared it with growth rates for the population in the middle of the 20th Century. This is an estimate of the birth rate and the death rate in the year 1000, birth rate roughly 70, death rate about 69.5. Remember when there's a growth rate of 1 percent, very much like your money, a growth rate of 1 percent leads to a doubling time at about 69 to 70 years. It's the same thing with humans. With a 1 percent growth rate, the population doubles in about 69 years. If you have the growth rate — if you double the growth rate, you have the time it takes for the population to double, so it's nothing more than the difference between the birth rate and the death rate to generate the growth rate. And here you can see in 1900, the growth rate was about 2 percent, which meant the doubling time was about five years. During the 1950s at the height of the baby boom, the growth rate was about 3 percent, which means the doubling time was about 26 years. In the year 2000, we have birth rates of about 15 per thousand, deaths of about 10 per thousand, low mortality populations, which means the growth rate is about one half of 1 percent, which means it would take about 140 years for the population to double.

Well, if we achieved immortality today, in other words, if the death rate went down to zero, then the growth rate would be defined by the birth rate. The birth rate would be about 15 per thousand, which means the doubling time would be 53 years, and more realistically, if we achieved immortality, we might anticipate a reduction in the birth rate to roughly ten per thousand, in which case the doubling time would be about 80 years. The bottom line is, is that if we achieved immortality today, the growth rate of the population would be less than what we observed during the post World War II baby boom.