### 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 numbers.

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:

• 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.
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.