the reals and models that are larger. Some of the latter are studied in non-standard analysis. Role in mathematical logic Deductive systems and completeness A deductive system consists of a set of logical axioms, a set of non-logical axioms, and a set of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas , that is, for any statement that is a logical consequence of there actually exists a deduction of the statement from . This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system. Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms. There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another. Further discussion Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously, there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, and modern algebra was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent. See also Axiomatic system Dogma First principle, axiom in science and philosophy List of axioms Model theory Regulæ Juris Theorem Presupposition Physical law Principle Notes References Further reading Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks. External links