Sentence:  If \alpha is an <WSD>ordinal number</WSD>, then by definition any two well-ordered sets of type \alpha are similar, i.e., there exists a one-to-one mapping from one set to the other. Consequently these sets have the same cardinality. Consequently to any <WSD>ordinal number</WSD> \alpha we may assign a cardinal number, the common cardinality of all well-ordered sets of type \alpha. 
Meaning of the word marked with <WSD> tags:  Such a number generalised to correspond to any cardinal number (the size of some set); "formally", the order type of some well-ordered set of some cardinality "a", which represents an equivalence class of well-ordered sets (exactly those of cardinality "a") under the equivalence relation "existence of an order-preserving bijection".