We now consider a third realisation of Numbers which we will call , defined as follows:
We can see that this is a similar algebra to |Num|; in fact, intuitively it seems to have the same meaning. We have simply replaced the usual Arabic characters with Roman numerals. This makes the arithmetic seriously difficult beyond very small numbers, but this is not our concern at this level; we should still expect our results to `match'. If we had used the Greek representation instead of Roman (they also used their alphabetic characters in a similar way) we should have the same expectations of the algebra. When algebras are related in this way -- where the difference between them is the choice of labels for the elements of the carrier sets, and the application of the operators gives corresponding results -- they are said to be isomorphic.
An isomorphism is a 1-1 homomorphism; so if g is a homomorphism from |Num| to then we can find another homomorphism h from to |Num| such that when we compose h with g we `get back to where we started'. Clearly, then, if two algebras are isomorphic their corresponding carriers will have the same cardinality.