Fullständighetsatsen

Fullständighetssatsen, often translated as the Completeness Theorem, is a fundamental result in mathematical logic. It establishes a crucial link between the syntactic and semantic notions of truth in formal logical systems. Specifically, for first-order logic, the theorem states that a formula is logically valid if and only if it is provable within the formal system. In simpler terms, any statement that is true in all possible interpretations (logically valid) can be formally derived using the rules of inference of the system. Conversely, any statement that can be formally derived is guaranteed to be true in all interpretations. This means that the proof system of first-order logic is sound (only valid formulas can be proven) and complete (all valid formulas can be proven). Kurt Gödel proved this theorem in 1929. The Fullständighetssatsen is vital for understanding the expressive power and limitations of formal logical systems, underpinning much of modern logic and theoretical computer science. It assures us that our formal deduction mechanisms are adequate for capturing all logical truths within first-order logic.

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