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I'm studying for a logic exam and I've become stuck on this problem. My instinct is that it isn't possible but I don't really know how to approach proving why,

Is it possible to express the following in $L$ using a finite set of sentences: $R(x, y)$ is a linear order with left endpoint $b$ and right endpoint $c$ and there are a finite number of distinct objects between $b$ and $c$ in this order?

Taroccoesbrocco
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Jonahhill
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You can express the property of being a linear order with certain endpoints, but, as the comment of Eric Wofsey suggests, it is not possible to write a finite set of formulas that says something like "there are finitely many distinct elements such that...". This follows from the Compactness Theorem: if you have a set of formulas that have models of arbitrarily large finite cardinality then you have an infinite model of that set of formulas.

Manlio
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    This correctly answers the question, which was about expressibility by a finite set of sentences, but it may be worth noting that the property in question cannot be expressed by any set of first-order sentences, not even by an infinite set. – Andreas Blass May 14 '18 at 21:52
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    Being somewhat picky - You can express it as $(\exists A)[ |A| < \omega \land (\forall z)[ b\prec < z \prec c \to z \in A] ]$, which is a first-order sentence. It is vital in this kind of problem to specify exactly which language is intended. The question almost certainly intended the language of linear orders, of course, but the important point is that it is not only the property, but also the language, that limits what can be expressed. – Carl Mummert May 15 '18 at 01:00