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topynate comments on Exponentiation goes wrong first - Less Wrong
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Aren't you mixing up "having a model" in the sense of knowing and thinking about one, and having one in the sense of one existing? An untutored person may have no real opinion on the relationship between his words and reality, but one nevertheless exists.
You would have to tell me what you mean by "existing." There is a real (though often informal) distinction between constructive and non-constructive proofs, and this distinction exists in model theory also.