In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, an amorphous set is an
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
which is not the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of two infinite
subsets.
[.]
Existence
Amorphous sets cannot exist if the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is assumed.
Fraenkel constructed a permutation model of
Zermelo–Fraenkel with Atoms in which the set of atoms is an amorphous set. After Cohen's initial work on forcing in 1963, proofs of the consistency of amorphous sets with
Zermelo–Fraenkel were obtained.
Additional properties
Every amorphous set is
Dedekind-finite
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' ont ...
, meaning that it has no
bijection to a proper subset of itself. To see this, suppose that
is a set that does have a bijection
to a proper subset. For each natural number
define
to be the set of elements that belong to the image of the
-fold
composition of with itself but not to the image of the
-fold composition.
Then each
is non-empty, so the union of the sets
with even indices would be an infinite set whose complement in
is also infinite, showing that
cannot be amorphous. However, the converse is not necessarily true: it is consistent for there to exist infinite Dedekind-finite sets that are not amorphous.
[.]
No amorphous set can be
linearly ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
.
[. In particular this is the combination of the implications which de la Cruz et al. credit respectively to and .] Because the image of an amorphous set is itself either amorphous or finite, it follows that every function from an amorphous set to a linearly ordered set has only a finite image.
The
cofinite filter on an amorphous set is an
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
. This is because the complement of each infinite subset must not be infinite, so every subset is either finite or cofinite.
Variations
If
is a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of an amorphous set into finite subsets, then there must be exactly one integer
such that
has infinitely many subsets of size
; for, if every size was used finitely many times, or if more than one size was used infinitely many times, this information could be used to coarsen the partition and split
into two infinite subsets. If an amorphous set has the additional property that, for every partition
,
, then it is called strictly amorphous or strongly amorphous, and if there is a finite upper bound on
then the set is called bounded amorphous. It is consistent with ZF that amorphous sets exist and are all bounded, or that they exist and are all unbounded.
References
{{Set theory
Axiom of choice
Cardinal numbers