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110 Mathematische Annalen we make use of the so-called "ordinal types" whose theory we have to set forth in the following paragraphs.

§7 The Ordinal Types of Simply Ordered Aggregates

We call an aggregate $$M$$ "simply ordered" if a definite "order of precedence" (Rangordnung) rules over its elements $$m$$, so that, of every two elements $$m_1$$ and $$m_2$$, one takes the "lower" and the other the "higher" rank, and so that, if of three elements $$m_1$$, $$m_2$$, and $$m_3$$, $$m_1$$, say, is of lower rank than $$m_2$$, and $$m_2$$ is of lower rank than $$m_3$$, then $$m_1$$ is of lower rank than $$m_3$$. The relation of two elements $$m_1$$ and $$m_2$$, in which $$m_1$$ has the lower rank in the given order of precedence and $$m_2$$ the higher, is expressed by the formulæ: (1)

Thus, for example, every aggregate $$P$$ of points defined on a straight line is a simply ordered aggregate if, of every two points $$p_1$$ and $$p_2$$ belonging to it, that one whose co-ordinate (an origin and a positive direction having been fixed upon) is the lesser is given the lower rank.

It is evident that one and the same aggregate can be "simply ordered" according to the most different laws. Thus, for example, with the aggregate $$R$$ of