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2.2: Theories and Their Models
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2.2.1: Introduction
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The axiomatic method and logic were made for each other. Formal logic provides the tools for formulating axiomatic theories, for proving theorems from the axioms of the theory in a precisely specified way, for studying the properties of all systems satisfying the axioms in a systematic way.
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2.2.2: Expressing Properties of Structures
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It is often useful and important to express conditions on functions and relations, or more generally, that the functions and relations in a structure satisfy these conditions.
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2.2.3: Examples of First-Order Theories
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Mathematics yields many examples of theories, e.g., the theories of linear orders, of groups, or theories of arithmetic, e.g., the theory axiomatized by Peano’s axioms.
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2.2.4: Expressing Relations in a Structure
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One main use formulas can be put to is to express properties and relations in a structure \(M\) in terms of the primitives of the language \(\mathcal L\) of \(M\).
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2.2.5: The Theory of Sets
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Almost all of mathematics can be developed in the theory of sets.
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2.2.6: Expressing the Size of Structures
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There are some properties of structures we can express even without using the non-logical symbols of a language. For instance, there are sentences which are true in a structure iff the domain of the structure has at least, at most, or exactly a certain number \(n\) of elements.
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2.2.7: Summary
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