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Infinity

Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of $\omega$(the mathematical symbol for the set of all integers)? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on an other worldly absolute truth. This idea has its origins in the Platonic concept of an ideal and perfect world of which the physical world is a dim reflection. Many mathematicians hold to this position in some form. They believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that all that exists in the physical world is imperfect and falls short of what we can apprehend with mathematical thinking.

For example all physical circles have imperfections, but the geometric circle is perfect. One can argue that this distinction no longer exists. We cannot construct a perfect circle, but we can construct computer programs that will describe the perfect circle to any desired degree of accuracy. A computer program can be interpreted by a physical device. Its instructions are usually carried out with absolute perfection. Computers can and do make errors but the probability of an error is small and there are techniques that can make that probability arbitrarily small. Thus we have something physical that comes close to being a Platonic ideal.

The infinite used to be thought of as a potential that is never fully realized. This perspective fell into disfavor as mathematicians constructed a hierarchy of infinite sets. that greatly extend mathematics. The practical consequences of these extension suggested there was some special mathematical insight into the infinite.

The hierarchy of infinite sets comes in two flavors, the ordinals and the cardinals. The ordinals generalize the construction of the integers. We construct the successor of $\omega$ just as we construct the successor of a finite integer. We take the union of $\omega$ and all the members of $\omega$. Ordinals are important because one can do induction on them in more powerful ways than one can do induction on the integers. It is induction up to particular ordinals that allow us to solve particular Halting Problems. For every Halting Problem there is some ordinal large enough to solve it.


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