amorphous set

Amorphous sets cannot exist if the axiom of choice is assumed. Fraenkel constructed a permutation model of Zermelo–Fraenkel with Atoms in which the set of atoms is an amorphous set. This is already sufficient for proving the consistency of the existence of an amorphous set with Zermelo–Fraenkel set theory with atoms.{{citation |title=The axiom of choice |last=Jech |first=Thomas J. |date=2008 |publisher=Dover Publications |isbn=978-0486318257 |location=Mineola, N.Y. |oclc=761390829}} After Cohen's initial work on forcing in 1963, proofs of the consistency of amorphous sets with Zermelo–Fraenkel set theory were obtained.{{citation |last=Plotkin|first=Jacob Manuel |date=November 1969|title=Generic Embeddings |url=https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/generic-embeddings/BEC36DA66935B30C082FDFB08F31D8D3 |journal=The Journal of Symbolic Logic |language=en |volume=34 |issue=3 |pages=388–394 |doi=10.2307/2270904 |jstor=2270904 |issn=0022-4812 |mr=252211|s2cid=250347797 }}

Every amorphous set is Dedekind-finite, meaning that it has no bijection to a proper subset of itself. To see this, suppose that $S$ is a set that does have a bijection $f$ to a proper subset. For each natural number $i\backslash ge\; 0$

define $S\_i$ to be the set of elements that belong to the image of the $i$-fold Iterated function but not to the image of the $(i+1)$-fold composition.

Then each $S\_i$ is non-empty, so the union of the sets $S\_i$ with even indices would be an infinite set whose complement in $S$ is also infinite, showing that $S$ 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.{{citation|last=Lévy|first=A.|title=The independence of various definitions of finiteness|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf|year=1958|author-link=Azriel Lévy|journal=Fundamenta Mathematicae|volume=46|pages=1–13|doi=10.4064/fm-46-1-1-13 |mr=0098671}}.

No amorphous set can be linearly ordered.{{citation

| last = Truss | first = John

| issue = 3

| journal = Fundamenta Mathematicae

| mr = 0469760

| pages = 187–208

| title = Classes of Dedekind finite cardinals

| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm84/fm84119.pdf

| volume = 84

| year = 1974| doi = 10.4064/fm-84-3-187-208

}}.{{citation

| last1 = de la Cruz | first1 = Omar

| last2 = Dzhafarov | first2 = Damir D.

| last3 = Hall | first3 = Eric J.

| doi = 10.4064/fm189-2-5

| issue = 2

| journal = Fundamenta Mathematicae

| mr = 2214576

| pages = 155–172

| title = Definitions of finiteness based on order properties

| url = http://h.web.umkc.edu/halle/relfin/orderfinite-revisions.pdf

| volume = 189

| year = 2006| doi-access = free

}}. In particular this is the combination of the implications $\backslash text\{Ia\}\backslash Rightarrow\backslash text\{II\}\backslash Rightarrow\backslash Delta\_3$ which de la Cruz et al. credit respectively to {{harvtxt|Lévy|1958}} and {{harvtxt|Truss|1974}}. 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. This is because the complement of each infinite subset must not be infinite, so every subset is either finite or cofinite.

If $\backslash Pi$ is a partition of an amorphous set into finite subsets, then there must be exactly one integer $n(\backslash Pi)$ such that $\backslash Pi$ has infinitely many subsets of size $n$; 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 $\backslash Pi$ into two infinite subsets. If an amorphous set has the additional property that, for every partition $\backslash Pi$, $n(\backslash Pi)=1$, then it is called strictly amorphous or strongly amorphous, and if there is a finite upper bound on $n(\backslash Pi)$ 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.

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{{Set theory}}

Category:Axiom of choice

Category:Cardinal numbers