Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

MORE ABOUT LIMITS 267
coincide. Of relations as a subject I have already
repeatedly indicated the immensity and the first-rate
importance. At present, I am asking you to consider
only so much of it as is necessary and sufficient for our
present purpose, which is that of preparing us to understand certain highly important meanings of the term
"limit."
Relations are endless in number and in variety and
they are omnipresent as well in practical life as in abstract
thought. There is one variety (including a vast multitude
of sub-varieties) to which I am going now to ask your best
attention. Before defining it, it will be helpful to consider a simple specific example of it. The example I am
going to use is one of the relations instanced a moment
ago. I mean the relation determined by the propositional
function: x is a positive integer less than a positive integer
y. Let us denote the relation by P. Observe what P is.
It is the class of couples: (i, 2), (i, 3), (I, 4), (I, 5),.;
(2, 3), (2, 4), (2, 5), * * *; (3, 4), (3, 5),;...;...;
and so on endlessly. Note that P has a field-the class of
all the positive integers. The relation P has numerous
properties; let me ask you to inspect very carefully just
three of them. The three are these: (a) if n be in P's
field, (n, n) is not a constituent of P,-that is, nPn is a
false proposition,-that is, P is not a relation which, like
identity, holds between a term and that same term;
(b) if n and n' are in P's field, then either (n, n') or else
(n', n) is a constituent of P-that is, nPn' or else n'Pn;
(c) if nPn' and n'Pn", then also nPn". Because the relation has these three properties, it is called a serial relation,
or a series, or a sequence, or a specimen of linear order.
You detect at once how to define these equivalent terms.
The definition is as follows: A serial relation (or series or