Next: Cardinal numbers Up: Mathematical structure Previous: Axiom of infinity Contents
PDF version
of this book
Next: Cardinal numbers Up: Mathematical
structure Previous: Axiom of infinity
Contents
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 
? 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. It is impossible to construct a perfect circle, but there are computer programs that will describe the perfect circle to any desired degree of accuracy. A computer program is 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 there are physical objects 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 extensions led many mathematicians to feel that reasoning about complex infinite sets could provide an intuitive window into useful mathematics that could not be developed in other more mundane ways.
The hierarchy of infinite sets comes in two flavors, the
ordinals and the cardinals. The ordinals
generalize the construction of the integers. The successor of is constructed like the
successor of a finite integer is constructed. It is the union of
and all the members of . 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. The ordinal numbers
are developed starting in Section 6.2. Loosely speaking ordinals
represent different ways of ordering operations. Cardinals
represent objects of different absolute size. Every integer is a
cardinal number.
PDF version
of this book
Next: Cardinal numbers Up: Mathematical
structure Previous: Axiom of infinity
Contents
| home | consulting | videos | book | QM FAQ | contact |