Scientists who speculate on the future of human longevity have a broad range of views ranging from the promise of immortality, to radical life extension, to declines in life expectancy. Among those who contend that radical life extension is already here, or on the horizon, or immortality is forthcoming, elements of their reasoning appear surprisingly close, if not identical, to a famous mathematical paradox posed by the ancient Greek mathematician known as Zeno. Here we examine the underlying assumptions behind the views that much longer life expectancies are forthcoming or have already arrived, and place their line of reasoning within the context of a new Zeno paradox described here as The Paradox of Immortality.