First, one extends the signature by adding a new constant symbol for every element of M . The complete theory of M for the extended signature σ' is called the elementary diagram of M . In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for any two distinct new constant symbols c and c' . Using the compactness theorem , the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly κ. It contains an isomorphic copy of M as an elementary substructure.