INFINITARY LOGIC Meaning and
Definition
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Infinitary logic is a branch of mathematical logic that extends the concepts and tools of classical logic to deal with infinitely long sentences or formulas. In classical logic, sentences are typically formed by combining a finite number of atomic sentences using logical connectives such as conjunction, disjunction, and implication. However, there are certain situations in mathematics and theoretical computer science where we need to reason about infinitely long structures or infinite collections of objects.
Infinitary logic provides a formal framework to handle these situations by allowing for infinite conjunctions and disjunctions, as well as infinitely long formulas. It introduces new connectives like the "infinitary conjunction" (∧ω) and "infinitary disjunction" (∨ω) to express that a statement holds for an infinite collection of objects or that an infinite number of conditions must simultaneously be satisfied.
The language of infinitary logic allows for flexible and precise representation of mathematical structures that are infinite or potentially infinite, such as sequences, sets, and functions. It has applications in various areas of mathematics, including set theory, model theory, and proof theory. Infinitary logic also serves as a foundation for the study of infinitary combinatorics, which deals with the counting and arrangement of infinite objects.
Overall, infinitary logic extends the expressive power of classical logic to handle infinite situations, providing a framework for reasoning and investigating mathematical structures that are unbounded in size or scope.
Etymology of INFINITARY LOGIC
The word "infinitary logic" is formed by combining two terms: "infinity" and "logic".
The term "infinity" derives from the Latin word "infinitas", which means "boundlessness". It is composed of two Latin components: "in" (not) and "finitas" (bounded). In mathematics and philosophy, "infinity" refers to a concept of endlessness or unboundedness.
The word "logic" comes from the Ancient Greek word "logikē", which means "the science of reasoning or thinking". It is derived from "logos", meaning "reason" or "word". Logic encompasses the rules and principles used in reasoning, thinking, and drawing valid conclusions.
