Lecture 3

Axiomatic Systems
Geometry - Arithmetic - Set Theory

We saw in the previous lecture that the Greeks were abile to reduce geometry to a set of axioms and postulates. They assembled a collection of increasingly complex results (theorems) which followed from these basic "truths." This approach to mathematics dominated mathematics for almost 2000 years.
Following this axiomatic approach to geometry, in 1889 Giuseppe Peano provided an axiomatic formulation of the natural numbers based on his Axioms. This generates the so called Peano Arithmetic.

Although the Peano axioms generate the natural numbers, there are theorems about natural numbers, which are unprovable via the Peano axioms (e.g. Goodstein's Theorem). This follows from Goedel's celebrated Incompleteness Theorem, which will be discussed next lecture.

David Hilbert, in 1900, formulated as his second problem, the Compatibility of the arithmetical axioms

Hilbert's sixth problem deals with the axiomitzation of physics!

Clearly, Hilbert was enamored of the process of axiomitization. Unfortunately, the process of reducing mathematical analysis to a formal set of axioms must always fail. There will always be statements which can be made which are independent of any finite set of axioms.

One of the most ambitious projects to reduce mathematics to a purely logical, formal axiomatic system was the work of Russell and Whitehead.

Russell and Whitehead sought, in their monumental work Principia Mathematica [published in 3 volumes in 1910, 1912, 1913],

"to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms", [Russel, My Philosophical Development, p. 57]
In fact, it took 362 pages to prove that 1+1=2 in their system!

Set theory was next to be put under the "axiomatic looking glass".

Naive Set Theory follows from intuitive definitions. Suppose we grant the truth of the following obvious "axiom"

If P(x) is a property, then {X | P(X)} is a set.
This lead's to Russel's paradox if P is the property "x is not an element of x". Consider
A = { X | X is not an element of X}
Is A and element of A. If it is, then A is an element of A. If it doesn't, then A is an element of A. Hence the contradiction.

To get around this problem, the notion of Axiomatic Set Theory was developed.

Z = Zermelo Set Theory was developed, followed by an extension to

ZF = Zermelo Frankel Set Theory and finally

ZFC = Zermelo Frankel Set Theory + Axiom of Choice

There are in fact other set theories which have been developed. Here is another reference to extensions of set theory

The Axiom of Choice and the Continuum Hypothesis are independent axioms in the sense that they can either be true or false within set theory.