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Power set axiom

The power set axiom is the last axiom of standard set theory.

\begin{displaymath}\forall x \exists y \forall z [ z \in y \equiv z \subseteq x ]\end{displaymath}

This says for every set $x$ there exists a set $y$ that contains all the subsets of $x$. $z$ is a subset of $x$ ($z \subseteq x$) if every element of $z$ is an element of $x$.

The axiom of the power set completes the axioms of ZF or Zermelo Frankel set theory. They are summarized in Figure 4.3.

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. We 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 4.4.

By Cantor's proof in Section 3.7 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 we 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 3 we use computers as 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.

Figure 4.3: The axioms of ZF set theory
\begin{figure} % latex2html id marker 778 \sf\begin{enumerate} \item Axiom of ex... ...[ z \in y \equiv z \subseteq x ]\end{displaymath}\par\end{enumerate}\end{figure}
Figure 4.4: Constructing a real from a subset of the integers
\begin{figure}\sf Following are the first few elements or a subset of the intege... ...e decimal. \end{enumerate}\par This generates $532.581219\ldots$\par\end{figure}

Completed second draft of this book

PDF version of this book
next up previous contents
Next: Trees of trees Up: Creative mathematics Previous: Searching all possible paths   Contents


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