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Consider a model of the empty unsorted signature. Equivalently, a model of the signature having a single sort, and no function or relation symbols. Intuitively, such a model should be called a "set."

However, the emphasis is all wrong. For example, in ZFC everything is a set, but I don't feel comfortable saying that everything in ZFC is a model of the empty signature. Firstly, because its just false. Secondly, because we should only talk about models up to isomorphism, however your average ZFC set is interesting beyond its cardinality; that is, beyond its structure up to isomorphism. We care about more than just the cardinality of your average ZFC set.

Is there a good word to mean "a model of the empty signature"?

goblin GONE
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  • Why do you think it is false that everything in ZFC (i.e., every set) is a model for the empty signature? Secondly, a model is usually defined over a set with perhaps additional entities such operations or relations. – Berci Nov 28 '13 at 00:53
  • Sometimes we say "a set with no additional structure". – Andrés E. Caicedo Nov 28 '13 at 00:55
  • @Berci, it depends how you implement things, but for example if you defined "model" to mean a particular kind of ordered pair, well ZFC proves that there exists $x$ such that for all $a,b$ we have $x \neq (a,b).$ – goblin GONE Nov 28 '13 at 00:56
  • Ok, then consider the trivial correspondence $S \mapsto (S,\emptyset)$ between sets and structures on empty signatures. – Berci Nov 28 '13 at 00:59

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I don't quite understand why you say that "it is just false". Following the definitions of model theory, a model of the empty signature is, precisely, a non-empty set. The fact that some non-empty sets may carry additions structure does not mean that we can't just ignore that structure and view it just as a set. This is done often. For instance, every vector space is in particular an abelian group by forgetting some of the structure. Forget all of the structure and all you have is the underlying set. To say that any non-empty set is a model of the empty signature does not mean that set does not have naturally occurring extra structure. It just means we choose to ignore it.

Ittay Weiss
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