20

17

in counting this collection is used in counting every
collection that this number is used in counting.
I might add a fifth proposition to express that there is an absolutely smallest
number; but I do not think that of much importance.

By manipulating those four graphs according
to the rules of graphs we could reach every theorem
of the mathematics of whole numbers. But the rules
of the transformation of graphs do not depend upon
what is written in the different spots. So long as they are
preserved the same or different, as the case may be, letters or
any other meaningless expressions might be substituted.
In particular, in place of “collections” any other universe
might be adopted, and in place of “is used in
counting” a general symbol for any one dyadic relation. If this
change were made in the four graphs, the result would
be that they would define the relations of ordinal numbers
to one another, except of course that they omit to
say that there is a smallest number; and in fact