The set of classical orderings of a field compatible with a given place from the field to the real numbers is known to be bijective with the set of homomorphisms from the value group of the place into the two element group. This fact is generalized here to the set of “generalized orders” compatible with an “extended absolute value,” i.e., an absolute value allowed to take the value ∞. The set o...

Introduction 2 0.1. Some theorems in mathematics with snappy model-theoretic proofs 2 1. Languages, structures, sentences and theories 2 1.1. Languages 2 1.2. Statements and Formulas 5 1.3. Satisfaction 6 1.4. Elementary equivalence 7 1.5. Theories 8 2. Big Theorems: Completeness, Compactness and Löwenheim-Skolem 9 2.1. The Completeness Theorem 9 2.2. Proof-theoretic consequences of the complet...

Introduction 1 0.1. Some theorems in mathematics with snappy model-theoretic proofs 1 1. Languages, structures, sentences and theories 1 1.1. Languages 1 1.2. Statements and Formulas 4 1.3. Satisfaction 5 1.4. Elementary equivalence 6 1.5. Theories 7 2. Big Theorems: Completeness, Compactness and Löwenheim-Skolem 9 2.1. The Completeness Theorem 9 2.2. Proof-theoretic consequences of the complet...

In view of Albert's classical result that the dimension of each ordered, associative proper algebra over its center is necessarily infinite (cf. [1]), it seems not unlikely that a similar statement also holds for the rank of each proper formally real quasifield F, i.e. for the dimension of F over its kernel. Indeed, for some classes of ordered near fields and for ordered quasifields admitting a...