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Need an elementary logic book that completely covers the completeness theorem (no pun intended).
Hurkyl said:Gödel proved a completeness theorem in addition to his two incompleteness theorems for logic. There are probably other 'completness theorem's too both in logic and in other contexts, so it's not clear that's the one the OP means.
If the OP does mean Gödel's completeness theorem, I imagine it should be in just about any good introductory text on formal logic. (i.e. a text meant to teach the discipline of formal logic, rather than an 'introduction to proofs in mathematics'-type book)
The Complete Completeness Theorem is a fundamental theorem in mathematical logic that states that any consistent set of first-order sentences can be satisfied by a model. In other words, if a set of sentences does not contradict each other, there exists a model that satisfies all of them.
The Complete Completeness Theorem is important because it guarantees the existence of a model for any consistent set of sentences in first-order logic. This allows us to prove the validity of arguments and theorems without having to check every single model, saving time and effort.
Completeness is a property of a logical system that describes its ability to prove all valid arguments. The Complete Completeness Theorem is a specific theorem that proves the completeness of a logical system.
The Complete Completeness Theorem is used in mathematics to prove the validity of arguments and theorems, particularly in the field of first-order logic. It also has applications in other areas of mathematics, such as set theory and model theory.
Yes, the Complete Completeness Theorem only applies to first-order logic and cannot be extended to more complex logical systems. It also assumes the existence of a countable model, which may not always be the case. Additionally, it does not guarantee a unique model for a consistent set of sentences.