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Mächtigkeit

Mächtigkeit is a German noun that denotes power, might, or strength, as well as the magnitude or size of something. In everyday language it refers to the capacity to influence or control events or situations, for example the political Mächtigkeit of a state or the economic Mächtigkeit of a corporation. In philosophy, political theory, and sociology the term is used to discuss power relations, legitimacy, and authority, often in relation to other forms of control.

In mathematics, particularly in set theory, Mächtigkeit means the cardinality or size of a set. The Mächtigkeit

Etymology traces Mächtigkeit to macht (power) and the suffix -igkeit, forming a noun that conveys the quality

of
a
finite
set
is
a
natural
number
equal
to
the
number
of
its
elements.
A
set
is
finite
exactly
when
it
has
a
finite
Mächtigkeit;
otherwise
it
is
infinite.
Infinite
Mächtigkeiten
are
studied
in
terms
of
their
relative
sizes:
some
infinite
sets
are
countable
(abzählbare
Mächtigkeit),
such
as
the
natural
numbers,
while
others
are
uncountable
(unabzählbare
Mächtigkeit),
such
as
the
real
numbers.
Cantor’s
theory
shows
that
infinite
Mächtigkeiten
can
be
ordered
by
cardinal
numbers,
with
ℵ0
(aleph-null)
being
the
Mächtigkeit
of
the
natural
numbers
and
higher
cardinals
for
larger
sets.
The
concept
is
central
to
discussions
of
computability,
analysis,
and
logic.
or
extent
of
power
or
size.
In
German,
it
is
a
formal
term
whose
usage
depends
on
context,
ranging
from
everyday
descriptions
of
influence
to
precise
mathematical
definitions.