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Maths › 03. Logic and Foundations ›

4.5 Examples

Model theory treats many familiar mathematical objects as structures for suitable first order languages, and this viewpoint allows groups, rings, fields, ordered sets, graphs, and geometric systems to be studied using a common formal framework. Algebraic Structures Algebraic structures are among the most natural examples of first order structures, because they are usually described by a domain together with operations and distinguished constants. A group may be represented in the...
logic model theory structures algebra geometry May 2, 2026
Maths › 03. Logic and Foundations ›

4. Structures and Models

Model theory studies mathematical structures through the formal languages used to describe them, and it asks how much information about a structure can be expressed by first order formulas. A structure consists of a domain together with interpretations of the symbols in a language, and this framework allows groups, rings, fields, ordered sets, graphs, and many other mathematical objects to be treated within one common logical setting. The chapter begins...
logic model theory structures models May 2, 2026
Maths › 03. Logic and Foundations ›

2.3 Structures

In first order logic, formulas do not have meaning by themselves, and meaning is provided by specifying a structure in which the symbols of the language are interpreted as actual objects, functions, and relations. Structures A structure consists of a domain together with interpretations of all symbols in the signature, and it serves as the semantic environment in which formulas are evaluated. Definition 2.19 (Structure) A structure $\mathcal{M}$ for a...
logic first order logic structures interpretations models May 2, 2026
Maths › 00. General Mathematics ›

01.4 Finite vs Infinite Objects

1.4 Finite vs Infinite Objects Mathematics distinguishes sharply between finite and infinite objects. Many definitions look similar in both settings, but their behavior diverges in essential ways. Understanding this boundary is necessary for reasoning, proof design, and algorithmic thinking. A set is finite if its elements can be listed completely and counted by a natural number. |A| = n for some n ∈ N A set is infinite if no...
mathematics foundations infinity cardinality structures May 2, 2026
Maths › 00. General Mathematics ›

01.1 Abstract Objects and Structures

1.1 Abstract Objects and Structures Mathematics studies objects by stripping away irrelevant detail. A number, a function, a graph, a space, and a group are not studied as physical things. They are studied through the rules they satisfy, the relations they support, and the transformations that preserve them. An abstract object is an object considered through its mathematical properties rather than its material form. The integer 3 can count apples,...
mathematics foundations abstraction structures May 2, 2026