Infinities

Two complementary distinctions:

• 1.Extendibility versus Divisibility
• 2.Actual versus Potential

	EXTENDIBILITY	DIVISIBILITY
POTENTIAL	Time [for Aristotle]	Finite Intervals [for Aristotle]
ACTUAL	Void [for the Atomists]	Finite Intervals [Zeno according to Aristotle]

Finite Intervals that are actually infinitely divisible would lead to continua composed from points. Aristotle rejected point-like continua on the grounds that such structures were incoherent.

For another example of the difference between potential and actual infinities, consider the natural numbers: 1, 2, 3, . . .. We can construe them as being potentially infinite or actually infinite.

As Potentially Infinite:For any number n, we can go on to a larger number n+1. The process is un-ending.

As Actually Infinite:Consider the collection as a whole. We represent collections by using curly brackets: {1, 2, 3, . . .}. How BIG is this collection? That is, how many elements does it contain? The collection contains an INFINITE NUMBER of elements. The problem for making sense of this notion is to make sense of the idea that an INFINITE NUMBER is really a NUMBER!