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

272 MATHEMATICAL PHILOSOPHY
be included in R' we say naturally that R is a part of R'.
I need hardly point out the fact that a relation includes
itself and is thus a part of itself. R and R' are identical
when and only when each is a part of the other. Now
suppose R to be a part of R' and suppose R' to be a
sequence (series); then R is also a sequence obviously;
and thus, as you see, one sequence may be a part of
another. It is plain that, if a sequence R be a part of a
sequence R', the field F of R is a part of the field F' of R';
and conversely that, if a class F (of two or more terms)
be a part of the field F' of a sequence R', then F is the
field of a sequence R included in R'. Consider, for
example, our familiar friend P; its F is the class of
positive integers; take any part of F,-any part containing two or more terms,-say, the class C composed of
2, 3 and 7; C is the field of the sequence composed of the
couples (2, 3), (2, 7), (3, 7); this sequence is a part of P.
It will be enlightening to notice that P is itself a part of
another sequence; let Q be the sequence determined by
the propositional function, x is a positive real number less
than a positive real number y; you see that Q is a sequence,
that its field is the class of all positive real numbers, that
this field includes the field of P; that every couple in P
is also in Q; and that P is a part of Q. You readily see
that if one sequence be a part of a second, and the
second a part of a third, the first is a part of the third.
When we speak of the (amount of) difference between
a term t in the field of a sequence. S and a term t', let it be
always understood that t' is either in the field of S or in
the field of a sequence including S.
I hope we are now prepared to grasp the following
definition of the term " limit."
D3: Let V be a variable whose range R is included in