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

270 MATHEMATICAL PHILOSOPHY
series P"',-the class of couples (x, y),-determined by the
propositional function: x is a real number (greater than
i) less than a real number y (less than 2); it is clear that
P"' is beginningless and endless-each term in its field
is both a predecessor and a successor. If t be a term in the
field of a sequence and be at once a successor of ti and a
predecessor of t2, then t is said to be between tl and t2. If
the sequence be S, we may say that the terms between
ti and t2 are S-intermediate to ti and t2. If between every
two terms in the field of a sequence there is a term of the
field, the sequence is said to be dense; thus, P"', for
example, is dense, while P, P', P" are not. You will
hardly confuse the notion of a dense sequence with that
of a dense difference kind k. The amounts of such a kind
constitute a field of a dense sequence but the differencekind is not itself a sequence.
It is noteworthy that, in dealing with a sequence,
mathematicians do not usually state explicitly a propositional function determining it, though it is always possible and often helpful to do so; neither do they usually
indicate explicitly (as above done in the case of P) the
class of couples constituting the sequence, though this,
too, can be done if desired. For example, if a mathematician wishes to invite attention to our sequence P, he
will ordinarily say: Consider the sequence
I, 2, 3, 4,..., n, n+I,...
He will probably talk as if the row of numbers were the
sequence, though it is not-the sequence being, as we have
seen, a certain class of couples; the numbers in the row
constitute the field of P but the field as such he will
probably not mention; he will speak of the numbers as
the terms of the sequence, though they are merely the