 If we did not have a sign for identity in the language, we would let the domain of \(M\) be the collection of new constants \(\{c_0, c_1, \ldots \}\). But as it is, there may be a sentence in the form \(c_{i}=c_{j}\), with \(i\ne j\), in \(\Gamma''\). If so, we cannot have both \(c_i\) and \(c_j\) in the domain of the interpretation (as they are distinct constants). So we define the domain \(d\) of \(M\) to be the set \(\{c_i\) | there is no \(j\lt i\) such that \(c_{i}=c_{j}\) is in \(\Gamma''\}\). In other words, a constant \(c_i\) is in the domain of \(M\) if \(\Gamma''\) does not declare it to be identical to an earlier constant in the list. Notice that for each new constant \(c_i\), there is exactly one \(j\le i\) such that \(c_j\) is in \(d\) and the sentence \(c_{i}=c_{j}\) is in \(\Gamma''\).