| 1. | Expanding the above definition as equations yields the following finite axiomatization.
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| 2. | Tarski's axioms are a first order axiomatization of Euclidean geometry.
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| 3. | Therefore, no formal system is a complete axiomatization of full number theory.
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| 4. | In that sense the above axioms constitute a finite axiomatization of regular expressions.
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| 5. | Kozen defined a Kleene algebra to be any model of this finite axiomatization.
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| 6. | One such axiomatization begins with the following axioms that describe a discrete ordered semiring.
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| 7. | For instance, C-star algebra provides an alternative axiomatization to probability theory.
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| 8. | This axiomatization is now known as von Neumann-Bernays-G�del set theory.
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| 9. | We fix some axiomatization of the predicate calculus : logical axioms and rules of inference.
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| 10. | The standard axiomatization of the natural numbers is named the Peano axioms in his honor.
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