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The theory of finite groups is the set of first-order statements in the language of groups that are true in all finite groups (there are plenty of infinite models of this theory). It is not completely trivial to find any such statement that is not true for all groups: one example is "given two elements of order 2, either they are conjugate or ...
The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic. For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum.
A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language)
Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of ...
The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true in all models is provable".
The theory of first-order Peano arithmetic seems consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete.
6.1 Example of first-order perturbation theory – ground-state energy of the quartic oscillator 6.2 Example of first- and second-order perturbation theory – quantum pendulum 6.3 Potential energy as a perturbation
This objective required reformulating that geometry as a first-order theory. Tarski did so by positing a universe of points, with lower case letters denoting variables ranging over that universe. Equality is provided by the underlying logic (see First-order logic#Equality and its axioms). [2] Tarski then posited two primitive relations: