OF TRANSFINITE NUMBERS
101
with the cardinal number
. If the theorem is true for this aggregate, its truth for any other aggregate with the same cardinal number
follows at once by § 1. Let
be any part of
; we distinguish the following cases:
(a)
does not contain
as element, then
is either
[491] or a part of
, and so has as cardinal number either
or one of the numbers
, because we supposed our theorem true for the aggregate
, with the cardinal number
,
(b)
consists of the single element
, then
.
(c)
consists of
and an aggregate
, so that
.
is a part of
and has therefore by supposition as cardinal number one of the numbers
. But now
, and thus the cardinal number of
is one of the numbers
.
Proof of A.—Every one of the aggregates which we have denoted by
has the property of not being equivalent to any of its parts. For if we suppose that this is so as far as a certain
, it follows from the theorem D that it is so for the immediately following number
. For
, we recognize at once that the aggregate
is not equivalent to any of its parts, which are here
and
. Consider, now, any two numbers
and
of the series
; then, if
is the earlier and
the later,
is a part of
. Thus
and