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In an interpretation, are the domain and the subsets slated to go into the predicate letters, supposed to be well-defined sets? If the Axiom of Replacement is used to define a subset of the domain, using a formula, and this formula is undecidable for some $x$ in the domain, then I can't understand how a truth value can be assigned to the sentence "$x$ is in that subset" in the interpretation. I am very confused! Can anyone help?

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    http://mathoverflow.net/questions/72016/model-theory-confusion-closed2011-08-03
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    You should view the definition of model, and definition of truth in a model, as being like any other mathematical definition, in, say, group theory. And we can know what it *means* for a sentence to be true in a structure, even if we don't know *whether* the sentence is true in that structure. It is best not to drag formal set theory into the game. Model Theory is a field of mathematics like any other.2011-08-03
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    I would like to record a vote *against* closing. This, to me, appears to be a question about how model theory should be carried out inside an axiomatic set theory, which is a fair enough question. I'm no expert, but I suspect there may have been some confusion between theory and metatheory. I would certainly like to see a detailed answer from our resident set theorists.2011-08-04

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