axiom of infinity

The Axiom of Infinity simply asserts that an infinite set (a completed object you can build up from nothing) exists. This proposition must be introduced as an axiom because it cannot be proven from the other axioms of set theory. Since there are many ways to define infiniteness, The axiom is formally stated in various ways in various axiom systems (notation explained here):

• The Zermelo-Fraenkel axiom system states that there is a set W such that
  
  0 ∈ W & x ∈ W -> [x] e W
  
  That is, if x is an element of W, the set containing only x is also in W. It also seeds things by stating the empty set is in W
  
  
.
• The modified axiom system defined by Paul Bernays, having already defined "finiteness" in terms of ordinals while developing general set theory, builds upon his earlier definitions and states that the class of all natural numbers (the finite ordinals 0, 1, 2, and so on) is represented by a set, which is labeled with the Greek letter omega (ω):
  
  a ∈ ω <-> Nu(a)
  
  
• Another way to assert the existence of an infinite set is to define an infinite set as the domain of a function that puts the set into a one-to-one correspondence with a proper subset of itself, and then to state that at least one such function exists:
  
  (Ex, y, A) (y ⊂ x & A ⊆ (x Χ y)* & (((a, b) ∈ z) <-> ((c, b) ∈ A -> a=c) & ((a, d) ∈ A -> b=d))

Although controversial, it's not difficult to accept the idea of holding an infinite number of objects in your hand.

Get a pen and draw a short piece of line on the palm of your hand. Label the left end of the line 0 and the right end of this 1.

               _-_
              |   |_-_
           _-_|   |   |
          |   |   |   |
          |   |   |   |_-_
          |   |   |   |   |
          |   |   |   |   |
          |   |   |   |   |
          |   |   |   |   |
 _        |   |   |   |   |
/ \       |           `   \
|  \      |                |
\   \     |                |
 \   \    / o------------o |
  \   \__/  0           1  |
   \                       /
    \                     |
     -_                   /
       -_                /
         -_             |

We can consider this to be the segment of the real number line between 0 and 1. This is also called the unit interval. We are holding all of the numbers greater than 0 and less than 1 in this interval. If you remove any finite number of points from the interval, there will still be points left over.

Actually, the real number line is extreme overkill Let's take a step back and just consider the fractions

1, 1/2, 1/3, 1/4, ...

that is, 1 divided by any natural number. There is a point on the unit interval for each of these fractions. You can actually place little x's on your handy dandy unit interval where these points lie. Go ahead, you don't have to be perfect.

Now of course you don't have enough time to mark them all off, but you can see they have to exist in the real number line. When you drew your line, you drew over each of these points. Now, one point exists for each natural number. If we consider our set of 1/x points complete, we have to consider the set of natural numbers complete, as was posed in the axiom.

Now go wash your hands. Although I suppose if you're a real math fanatic you can get it tattooed on.

Hang on a minute! Before you include the Axiom of Infinity in your next abstract system, there are two questions you need to ask yourself:

Do you want a system that is simple and powerful or consistent and complete?

Like many Mathematical axioms (best example: Euclid's fifth axiom), interesting things happen in systems where the axiom is denied. In particular, Kurt Gödel showed that systems including the Axiom of Infinity could not be proved consistent or complete. But these properties may be provable in finite systems¹.

Do you want a system that is simple or has some relation to reality?

It should be clear that the Axiom of Infinity is empirical in nature. As evidenced by the existential quantifier of predicate logic, allowing abstract systems to make empirical claims can lead to all sorts of ontological problems. Futhermore, an infinite universe may not jibe with your cosmology of choice (see The Universe is Finite).

1: Robert S. Wilson has written a great paper on this idea: http://www.sonoma.edu/People/SWilson/Papers/finite/