, i.e., addition is associative. , i.e., addition is commutative. , i.e., multiplication is associative. , i.e., multiplication is commutative. , i.e., multiplication distributes over addition. , i.e., zero is an identity for addition, and an absorbing element for multiplication (actually superfluous). , i.e., one is an identity for multiplication. , i.e., the '<' operator is transitive. , i.e., the '<' operator is irreflexive. , i.e., the ordering satisfies trichotomy. , i.e. the ordering is preserved under addition of the same element. , i.e. the ordering is preserved under multiplication by the same positive element. , i.e. given any two distinct elements, the larger is the smaller plus another element. , i.e. zero and one are distinct and there is no element between them. In other words, 0 is covered by 1, which suggests that natural numbers are discrete. , i.e. zero is the minimum element. The theory defined by these axioms is known as PA−; the theory PA is obtained by adding the first-order induction schema. An important property of PA− is that any structure satisfying this theory has an initial segment (ordered by ) isomorphic to . Elements in that segment are called standard elements, while other elements are called nonstandard elements. Undecidability and incompleteness According to Gödel's incompleteness theorems, the theory of PA (if consistent) is incomplete. Consequently, there are sentences of first-order logic (FOL) that are true in the standard model of PA but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as Robinson arithmetic. Closely related to the above incompleteness result (via Gödel's completeness theorem for FOL) it follows that there is no algorithm for deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence, PA is an example of an undecidable theory. Undecidability arises already for the existential sentences of PA, due to the negative answer to Hilbert's tenth problem, whose proof implies that all computably enumerable sets are diophantine sets, and thus definable by existentially quantified formulas (with free variables) of PA. Formulas of PA with higher quantifier rank (more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of the arithmetical hierarchy. Nonstandard models Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies