In the context of Wikipedia, a "stub" is a small, incomplete article that provides some basic information about a topic but lacks detailed content. A "Mathematical logic stub" refers specifically to a brief article related to the field of mathematical logic that needs further expansion and development. Mathematical logic itself is a subfield of mathematics and philosophy that focuses on formal systems, proof theory, model theory, set theory, and computability, among other areas.

Mathematical logicians are scholars and researchers who study mathematical logic, a subfield of mathematics that focuses on formal systems, proofs, and the foundational aspects of mathematics. Their work lies at the intersection of mathematics, philosophy, and computer science, and it involves the exploration of various logical systems, including propositional logic, predicate logic, modal logic, and more.

Logical positivism, also known as logical empiricism, is a philosophical movement that developed in the early 20th century, primarily in the context of the Vienna Circle and the work of philosophers such as Moritz Schlick, Rudolf Carnap, and A.J. Ayer. It sought to synthesize elements of empiricism and formal logic, emphasizing the importance of scientific knowledge and the use of logical analysis in philosophical inquiry.

Logical truth refers to statements or propositions that are true in all possible interpretations or under all conceivable circumstances. In formal terms, a logical truth is typically a statement that can be proven to be true through logical deduction and does not depend on any specific facts or empirical evidence. One classic example of a logical truth is the statement "If it is raining, then it is raining." This statement is true regardless of whether or not it is actually raining because it holds true based solely on its logical structure.

Mathematical axioms are fundamental statements or propositions that are accepted without proof as the starting point for further reasoning and arguments within a mathematical framework. They serve as the foundational building blocks from which theorems and other mathematical truths are derived. Axioms are thought to be self-evident truths, although their acceptance may vary depending on the mathematical system in question.

Mathematical logic hierarchies refer to the structured classifications of various logical systems, mathematical theories, and their properties. These hierarchies help to categorize and understand the relationships and complexities between different logical frameworks.

Mathematical logic organizations are professional associations, societies, or groups that focus on the advancement and dissemination of research in mathematical logic and related areas. These organizations foster collaboration among researchers, provide platforms for sharing ideas, and often organize conferences, workshops, and publications in the field of mathematical logic.

In graph theory, an **infinite graph** is a graph that has an infinite number of vertices, edges, or both. Unlike finite graphs, which have a limited number of vertices and edges, infinite graphs can be more complex and often require different techniques for analysis and study. ### Key Characteristics of Infinite Graphs: 1. **Infinite Vertices**: An infinite graph can have an infinite number of vertices.

Structuralism in the philosophy of mathematics is an approach that emphasizes the study of mathematical structures rather than the individual objects that make up those structures. This perspective focuses on the relationships and interconnections among mathematical entities, suggesting that mathematical truths depend not on the objects themselves, but on the structures that relate them. Key aspects of mathematical structuralism include: 1. **Structures over Objects**: Structuralism posits that mathematics is primarily concerned with the relationships and structures that can be formed from mathematical entities.