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# Power set axiom

The power set axiom says the set of all subsets of any set exists. This is not needed for finite sets, but it is essential to define the set of all subsets of the integers.

This says for every set there exists a set that contains all the subsets of . is a subset of () if every element of is an element of .

The axiom of the power set completes the axioms of ZF or Zermelo Frankel set theory. They are summarized in Figure 6.4 that also includes the axiom of choice described in the next section.

From the power set axiom one can conclude that the set of all subsets of the integers exists. From this set on can construct the real numbers. One approach is to use the natural order of integers in the subset. The smallest member defines the integer part of the real. Successive integers form successive digits in the fractional part. Each fractional digit is between 0 and 9. Take the remainder from dividing the integer by 10 to get a digit between 0 and 9. Applying this process to every subset of the integers generates every real number many times. This procedure is illustrated in Figure 6.3.

By Cantor's proof in Section 5.7 one cannot map the reals onto the integers and by the above construction the same applies to maps between the integers and all subsets of the integers.

Constructions like those outlined in this and the previous section show how natural it can be to talk about infinite processes operating on infinite and even uncountable sets. This is part of the reason mathematicians think of the infinite as if it were a physical reality. Such arguments are logically sound and can be important mathematically.

The problem is that taking this approach too literally leads to a false intuition about the nature of infinity. It seems as if the mind is grasping the infinite when it is actually arguing about recursive processes carried out in a potentially infinite universe. In the next section I touch on an alternative direction for extending mathematics that grows out of an acceptance of the infinite as a potential that can never be actualized. It is a philosophical view that sees mathematical creativity and all creativity as a divergent and not a convergent process.

As mentioned at the beginning of Chapter 5 Computers are an accurate metaphor for a logically determined sequence of events. There is nothing magical about computers. It is the laws of physics that allow us to build computers and imply a universe in which logic is important in predicting future events. Whether or not there is fundamental randomness in the laws of physics is an open question that will be explored in the next two chapters.