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The theory of real closed fields is the theory in which the primitive operations are multiplication and addition; this implies that, in this theory, the only numbers that can be defined are the real algebraic numbers. As proven by Tarski, this theory is decidable; see Tarski–Seidenberg theorem and Quantifier elimination.
Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms.
The theory of the natural numbers with a successor function is complete and decidable, and is κ-categorical for uncountable κ but not for countable κ. Presburger arithmetic is the theory of the natural numbers under addition, with signature consisting of a constant 0, a unary function S, and a binary function +. It is complete and decidable.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
Even the real version of Schanuel's conjecture is not a necessary condition for the decidability of the theory. In their paper, Macintyre and Wilkie showed that an equivalent result to the decidability of Th ( R exp ) {\displaystyle \operatorname {Th} (\mathbb {R} _{\exp })} is what they dubbed the weak Schanuel's conjecture.
The elementary diagram of A is the set eldiag A of all ... it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for ...
Tarski's axiomatization, which is a second-order theory, can be seen as a version of the more usual definition of real numbers as the unique Dedekind-complete ordered field; it is however made much more concise by avoiding multiplication altogether and using unorthodox variants of standard algebraic axioms and other subtle tricks. Tarski did ...
Pages for logged out editors learn more. Contributions; Talk; Decidability of the first-order theory of the real numbers
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