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How does one show uniqueness of structures? If I have two structures that have certain properties, is there a general method for showing that they must be the same? Is this up to concept usually thought of as "with respect to isomorphism"?

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Of course, up to isomorphism. You need to build an isomorphism between two such structures. – Berci Jan 27 '13 at 0:16
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Of course this concept is thought of with "up to isomorphism", because given two sets of the same cardinality, every structure on the first would have an isomorphic structure on the latter.

You can restrict things a bit if you insist on fixing the underlying set, and if you insist that some additional (external) conditions be satisfied. For example $\{0,1\}$ is the smallest field of two elements whose elements are the least natural numbers. It's not an insightful example, but it's one nonetheless.

But your notion is close enough, if two structures are isomorphic then they are essentially the same. Be careful, though, to make sure that this depends greatly on the language and its interpretation. In the language of ordered sets clearly $\mathbb Q$ and $\mathbb Z$ (with their usual orders) are not isomorphic, but if we interpret the orders as empty instead then these are just two countable sets and they are isomorphic again.

To finish things up, let me actually answer the question, you need to show that any structure satisfying such and such property would have an isomorphism with your given structure.

For example, any infinite linearly ordered set that every element has only finitely many smaller elements is isomorphic to the natural numbers with their usual order. In this case you can even show that the isomorphism is unique.

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