definitely, and when we are asked
by certain mathematicians to practise ourselves in such thoughts as that
for infinite series a proper part can be the equal of the whole, where
equality is defined through the establishment of one-one correspondence,
we are really merely informed that among the group of symbols used to
denote the concrete steps of an ever open counting process are groups of
symbols that can be used to indicate operations that are of the same
type as the given one in so far as the characteristic of being an open
series is concerned. If there were anywhere an infinity of things to
count, an unintelligible supposition, it would by no means be true that
any selection of things from that series would be the equivalent of all
things in the series, except in so far as equivalence meant that they
could be arranged in the same type of series as that from which they
were drawn.

Similarly the mathematical conception of the continuum is nothing but a
formulation of the manner in which the cuts of a line or the numbers of
a continuous series must be chosen so that there shall remain no
possible cut or number of which the choice is not indicated.
Correspondence is reached between elements of such series when the
corresponding elements can be reached by an identical process. It seems
to me, however, a mistake to _identify_ the number continuum with the
linear continuum, for the latter must include the irrational numbers,
whereas the irrational number can never represent a spatial position in
a series. For example, the sqrt{2} is by nature a decimal involving an
infinite, i.e., an ever increasing, number of digits to express it and,
by virtue of the infinity of these digits, they can never be looked upon
as all given. It is then truly a number, for it expresses a genuine
numerical operation, but it is not a position, for it cannot be a
determinate magnitude but merely a quantity approaching a determinate
magnitude as closely as one may please. That is, without its complete
expression, which would be analogous to the self-contradictory task of
finding a greatest cardinal number, there can be no cut in the line
which is symbolized by it. But the operations of translating algebraic
expressions into geometrical ones and vice versa (operations which are
so important in physical investigations) are facilitated by the notion
of a one to one correspondence between number and space.

When we pass to the transfinite numbers, w