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is it true that any consistent first-order theory has a model?

In case of affirmative response:

1) Is it the Godelian completeness proof?

2) Is there a standard strategy for constructing set-theoretic models for consistent first-order theories?

Tahnks.

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This is equivalent to the statement that every sentence which is true in every first order model is provable from a particular deductive system, which is the original statement of Godel's completeness theorem. Sometimes the converse, that every provable statement is true in every model(the Soundess theorem)is added to the statement of the completeness too. This converse is not equivalent to your question.

There is a canonical way to construct this model which is due to Leon Henkin.

An online source which outline the construction can be found here: http://www.cs.nmsu.edu/historical-projects/Projects/completeness.pdf

If you prefer a version which leaves fewer details to the reader, you could look at The book "Mathematical Logic" by Ebbinghaus, Flum, and Thomas.

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