Paradox

SISYPHUS’S ROCK AND ZENO’S PARADOXES 55in the off position, it will be on if the button is pressed anodd number of times and off if pressed an even number oftimes. Now suppose that Thomson manages to press thebutton an infinite number of times by making one jab inone minute, a second jab in the next half minute, a third inthe next quarter minute, and so on. At the end of the twominutes of jabbing is the lamp on or off? It cannot be onbecause Thomson never turned it on without also turningit off. Nor can it be off: for after first turning it on, he neverturned it off without also turning it on.The appearance of contradiction is a mirage generatedby the incompleteness of the supposition. Thomson’s instructionsonly specify what happens at 2 - ½ n-1 minutes, not thesecond minute itself. Consider a man who tells us that everynumber less than 1 is either fair or foul. In the sequence ½,¼, 8, . . . the first member is foul, the second fair, alternatingso that ½ n is foul if n is odd and fair if n is even (Bennacerraf1970). Now, is the limit of the sequence fair or foul? It cannotbe foul because there is a fair after every foul. But neither canit be fair because there is a foul after every fair. The dilemmais spurious. The instructions only cover the sequence, sonothing is implied about a number outside the sequence.Others suggest that the “paradox of the gods” cannot behandled by Cantor:A man decides to walk one mile from A to B. A god waitsin readiness to throw up a wall blocking the man’s furtheradvance when the man has traveled ½ mile. A second god(unknown to the first) waits in readiness to throw up awall of his own blocking the man’s further advance whenthe man has traveled ¼ mile. A third god . . . & c. adinfinitum. It is clear that this infinite sequence of mere