Counting Countable Models

In this chapter we will see that there are more countable models of \( \operatorname {\mathrm {Th}}(({\mathbb {N}},<))\) than countable models of \( \operatorname {\mathrm {Th}}(({\mathbb {N}},S))\) , where S is the successor relation. This result and its proof further illustrate the role of orderings in classification of structures. The material presented here also serves as a background for the next chapter, where it will be argued that despite its simplicity, \(({\mathbb {N}},S)\) is a fundamental structure of mathematics. A detailed analysis of the structure of nonstandard models of \( \operatorname {\mathrm {Th}}(({\mathbb {N}},S))\) and \( \operatorname {\mathrm {Th}}(({\mathbb {N}},<))\) is followed by a much briefer outline of \( \operatorname {\mathrm {Th}}(({\mathbb {N}},+))\) and in the last section we will discuss the diversity of countable models of the theory of the standard model of arithmetic \( \operatorname {\mathrm {Th}}(({\mathbb {N}},+,\cdot ))\) .