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all positive rational numbers $p/q$ (where $p$ and $q$ are relatively prime integers) which are greater than $0$ and less than $1$, there is, firstly, their "natural" order according to magnitude; then they can be arranged (and in this order we will denote the aggregate by $R_{0}$) so that, of two numbers $p_{1}/q_{1}$ and $p_{2}/q_{2}$ which the sums $p_{1}+q_{1}$ and $p_{2}+q_{2}$ have different values, that number for which the corresponding sum is less takes the lower rank, and, if $p_{1}+q_{1}=p_{2}+q_{2}$ then the smaller of the two rational numbers is the lower. [497] order of precedence, our aggregate, since to one and the same value of $p+q$ only a finite number of rational numbers $p/q$ belongs, evidently has the form

$R_{0}=(r_{1},r_{2},...,r_{\nu },...)=({\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},{\tfrac {2}{3}},{\tfrac {1}{5}},{\tfrac {1}{6}},{\tfrac {2}{5}},{\tfrac {3}{4}},...)$,

where

$r_{\nu }<r_{\nu +1}$.

Always, then, when we speak of a "simply ordered" aggregate $M$, we imagine laid down a definite order or precedence of its elements, in the sense explained above.

There are doubly, triply, $\nu$-ply and ${\mathfrak {a}}$-ply ordered aggregates, but for the present we will not consider them. So in what follows we will use the shorter expression "ordered aggregate" when we mean "simply ordered aggregate."

Every ordered aggregate $M$ has a definite "ordinal type," or more shortly a definite "type," which we will denote by