How can I prove the compactness theorem for sets of sentences?

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SUMMARY

The compactness theorem for sets of sentences states that a set of sentences T in a language L has a model if and only if every finite subset of T has a model. The proof's first direction is straightforward, as every model of T is also a model for every finite subset of T. The challenge lies in proving the opposite direction, which requires a deeper understanding of model theory. Consulting established logic textbooks for existing proofs is highly recommended for clarity and guidance.

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rainwyz0706
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An expression of the compactness theorem for sets of sentences is that: let T be a set of sentences in L. Then T has a model iff every finite subset of T has a model.
Could anyone give me some hints how to prove this?
The first direction is straightforward: every model of T is a model of every subset of T. But what about the opposite direction? Any help is appreciated!
 
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My advice would be to not try to reinvent the wheel, and study the proofs given in any logic textbook.
 

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