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