On Axiomatic and Genetic Construction of Mathematical Theories

  • S. S. Demidov
Part of the Synthese Library book series (SYLI, volume 145)


There are two methods of constructing mathematical theories: genetic and axiomatic. The first of them implies the construction of a theory through successive generalizations, on the basis of simple concepts established earlier. For instance, one can construct in this way the theory of real numbers proceeding from the basic unit: by applying a certain natural process (that of counting) one obtains from it the series of positive integers 1, 2, 3, 4, 5,…; introduces the so called arithmetic operations for them; then, by making use of these operations, introduces the negative numbers and zero; then fractions as pairs of integers; and, finally, defines the real number as a section or fundamental sequence.


Differential Calculus Axiomatic Theory XIXth Century Axiomatic Method Fundamental Sequence 
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© D. Reidel Publishing Company 1980

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  • S. S. Demidov

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