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

MORE ABOUT LIMITS 269
cessors-successors for R, or R successors; that, if xRy,
x is a predecessor of y and y is a successor of x-more
precisely, that x is an R predecessor of y and y is an R
successor of x. You see immediately that every term
in R's domain is an R predecessor, that every term in
R's codomain is an R, successor, and that every term in
R's field is either a predecessor or a successor and is
generally (not always) both. Thus in the case of our
example P, I is a predecessor but not a successor, while
every other integer in the field is both; on the other hand,
in the case of P', i is a successor but not a predecessor,
while every other positive integer is again both. If a
term t be an R predecessor but not an R successor, the
sequence R is commonly and conveniently said to have a
beginning t-to begin at t; thus P has a beginning, it
begins at i. If t be a successor but not a predecessor, the
sequence has an end t-it ends at t; thus P' has an end,
it ends at I; P is endless, P' is beginningless. A sequence
may have both beginning and end or neither. An
example of the former is the sequence P" determined by
the propositional function: x is a positive integer less than
a positive integer y not greater than Io; you see that
P" is a sequence and that it has a beginning, I, and an
end, Io. The field of P" is finite. Can a sequence
whose field is infinite have both beginning and end? Yes;
consider the sequence determined by the propositional
function: x is a real number (equal to or greater than I)
less than a real number y (not greater than 2); you see
that the relation determined by the function is a sequence,
that the sequence begins at I and ends at 2, and that the
field is infinite-the class whose terms are I, 2 and all the
intervening real numbers. For an example of a sequence
having neither beginning nor end, we may take the