Are there any complete and consistent theories?
Yes, there are many interesting complete and consistent
theories. Gödel's theorem applies only to theories that satisfy
"certain conditions" - essentially, that a certain amount of
arithmetic can be done in the theory. A theory for which these
conditions do not hold may be both complete and consistent, and
doesn't have to be at all trivial.
A good example is the elementary theory of the real field (also
known as the theory of real-closed fields). This theory has symbols 0
and 1, symbols for adding, multiplying, subtracting, dividing real
numbers, and statements built up from equalities and inequalities,
using propositional connectives and quantifiers over the real
Without going into details, the following examples of statements of
the theory should give a good idea of what can be said in the theory:
There is at least one potentially baffling aspect to the above
example. Since the natural numbers are a part of the real numbers, why
doesn't Gödel's theorem apply to the theory of the real field? The
answer is that even though the natural numbers are a part of the real
numbers, we can't define "natural number" using only the
language of the theory of the real field. Thus, for example, the
statement "there are natural numbers x,y,z greater than 0 such that
x5+y5=z5" can't be formulated in that
language. The (true) statement "there are real numbers x,y,z greater
than 0 such that x5+y5=z5" can.
- Every non-negative number has a square root.
- The equation x4+5x2-8=0 has exactly two real solutions.
- For every two real numbers x and y such that 0<x<y, there is a real
number z such that x<z2<y.