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One standard modern approach is to prove the Compactness Theorem using ultraproducts. I did so at the end of a short summer course on model theory: see

http://www.math.uga.edu/~pete/modeltheory2010Chapter6.pdfhttp://alpha.math.uga.edu/~pete/modeltheory2010Chapter6.pdf

One standard modern approach is to prove the Compactness Theorem using ultraproducts. I did so at the end of a short summer course on model theory: see

http://www.math.uga.edu/~pete/modeltheory2010Chapter6.pdf

One standard modern approach is to prove the Compactness Theorem using ultraproducts. I did so at the end of a short summer course on model theory: see

http://alpha.math.uga.edu/~pete/modeltheory2010Chapter6.pdf

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Pete L. Clark
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One standard modern approach is to prove the Compactness Theorem using ultraproducts. I did so at the end of a short summer course on model theory: see

http://www.math.uga.edu/~pete/modeltheory2010Chapter6.pdf