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The Löwenheim-Skolem Theorem asserts that if a first-order theory has an infinite model, then it has models of every infinite cardinality. This theorem highlights the limitations of first-order logic in characterizing the size of infinite structures, leading to the Skolem paradox where countable models can exist for uncountable theories.
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First-Order Logic (FOL) is a formal system used in mathematics, philosophy, linguistics, and computer science to express statements about objects and their relationships. It extends propositional logic by incorporating quantifiers and predicates, allowing for more expressive and detailed representations of knowledge and reasoning processes.
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Model theory is a branch of mathematical logic that studies the relationships between formal languages and their interpretations, or models. It provides tools to analyze the structure and properties of mathematical systems by examining the models that satisfy given sets of axioms or theories.
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Cardinality refers to the measure of the 'number of elements' in a set, which can be finite or infinite, and is crucial in understanding the size and comparison of sets in mathematics. It plays a fundamental role in set theory, enabling mathematicians to distinguish between different types of infinities and to explore properties of sets in various mathematical contexts.
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Uncountable sets are infinite sets that cannot be put into a one-to-one correspondence with the set of natural numbers, meaning their cardinality is strictly larger than that of countable sets. The most famous example of an uncountable set is the set of real numbers, which demonstrates that not all infinities are equal in size.
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Axiomatic set theory is a branch of mathematical logic that uses a formal system to define sets and their relationships, providing a foundation for much of modern mathematics. It addresses paradoxes and inconsistencies in naive set theory by introducing axioms that precisely dictate how sets can be constructed and manipulated.
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The Completeness Theorem, established by Kurt Gödel, states that every logically valid formula in first-order logic is provable, ensuring that the axioms and inference rules are sufficient to derive all truths expressible in the system. This theorem is fundamental in mathematical logic as it bridges the gap between semantic truth and syntactic provability, highlighting the power and limitations of formal systems.
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The Compactness Theorem in mathematical logic states that a set of first-order sentences has a model if and only if every finite subset of it has a model, highlighting the interplay between local consistency and global consistency. This theorem is fundamental in model theory, providing a powerful tool for proving the existence of models and for transferring properties from finite to inFinite structures.
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An elementary extension is a type of model in mathematical logic where a structure is expanded to a larger structure without changing the truth values of first-order statements. This ensures that the two structures are elementarily equivalent, meaning they satisfy the same first-order properties.
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A saturated model in model theory is one that realizes all types that are consistent with its theory, given its cardinality. This makes it a powerful tool for understanding the structure and properties of mathematical theories by providing a rich and comprehensive framework for analysis.
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Elementary equivalence is a relationship between two structures in model theory, indicating that they satisfy the same first-order sentences. This concept is crucial for understanding how different structures can be indistinguishable from the perspective of first-order logic, despite potentially differing in other logical or structural properties.
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