at the start of that hour. But since Aristotle holds that such treatments of infinity are impossible and ridiculous, the world cannot have existed for infinite time. The most sophisticated medieval arguments against an infinite past were later developed by the early Muslim philosopher, Al-Kindi (Alkindus); the Jewish philosopher, Saadia Gaon (Saadia ben Joseph); and the Muslim theologian, Al-Ghazali (Algazel). They developed two logical arguments against an infinite past, the first being the "argument from the impossibility of the existence of an actual infinite", which states: "An actual infinite cannot exist." "An infinite temporal regress of events is an actual infinite." "Thus an infinite temporal regress of events cannot exist." This argument depends on the (unproved) assertion that an actual infinite cannot exist; and that an infinite past implies an infinite succession of "events", a word not clearly defined. The second argument, the "argument from the impossibility of completing an actual infinite by successive addition", states: "An actual infinite cannot be completed by successive addition." "The temporal series of past events has been completed by successive addition." "Thus the temporal series of past events cannot be an actual infinite." The first statement states, correctly, that a finite (number) cannot be made into an infinite one by the finite addition of more finite numbers. The second skirts around this; the analogous idea in mathematics, that the (infinite) sequence of negative integers "..-3, -2, -1" may be extended by appending zero, then one, and so forth; is perfectly valid. Both arguments were adopted by later Christian philosophers and theologians, and the second argument in particular became more famous after it was adopted by Immanuel Kant in his thesis of the first antinomy concerning time. Modern revival Immanuel Kant's argument for temporal finitism, at least in one direction, from his First Antinomy, runs as follows: Modern mathematics generally incorporates infinity. For most purposes it is simply used as convenient; when considered more carefully it is incorporated, or not, according to whether the axiom of infinity is included. This is the mathematical concept of infinity; while this may provide useful analogies or ways of thinking about the physical world, it says nothing directly about the physical world. Georg Cantor recognized two different kinds of infinity. The first, used in calculus, he called the variable finite, or potential infinite, represented by the sign (known as the lemniscate), and the actual infinite, which Cantor called the