# Axiomatic Theories

Reference work entry

First Online:

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_420

## Abstract

One of the first steps in axiomatizing a theory is to list the primitive notions. A familiar example is the classical case of Euclidean geometry. We can take as primitives the following three notions: the notion of point, the notion of betweenness – one point being between two others in a line – and the notion of equidistance – (the distance between given points being the same as the distance between two other given points). Other geometric notions can then be defined in terms of these three notions. For example, the line generated by two distinct points *a* and *b* is defined as the *set* of all points *c* which are between *a* and *b*, which are such that *b* is between *a* and *c*, or which are such that *a* is between *c* and *b*.

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## References

- Bourbaki, N. 1950. The architecture of mathematics.
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*Vorlesungen über neuere Geometrie*. Leipzig: Springer.Google Scholar - Suppes, P. 1957.
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© Macmillan Publishers Ltd. 2018