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Assuming the Axiom of Choice, every cardinal is either finite (i.e., an element of $\omega$) or Dedekind-infinite (i.e., in bijection with a proper subset of itself). This dichotomy is not true in ZF, however; in particular, it is consistent with ZF that there exist sets which are non-finite but cannot be partitioned into two non-finite pieces. Such sets are called "amorphous," and models of ZF containing amorphous sets can be constructed by building a permutation model of ZFA (ZF with urelements) and applying the Jech-Sochor embedding theorem.

Naturally, there are lots of types of structure which amorphous sets simply cannot have. For example, no amorphous set can be linearly ordered, since a linear order on a non-finite set allows us to either inject $\omega$ into the set, or partition the set into a "left" and "right" piece, both of which are non-finite. However, amorphous sets can still have some structure. For example, say that a set is "even" if it can be written as a disjoint union of 2-element subsets. Then, again using permutation models, we can construct a model of ZF which contains an even amorphous set (the lack of choice prevents us from using the evenness of the set to partition it into two non-finite pieces).

My question is twofold. First, say that a set is "odd" if it can be written as the disjoint union of a singleton and an even set.

Question 1: Is ZF+"there exists an amorphous set"+"every set is either even or odd" consistent? (It occurs to me that this may depend on whether the above "or" is meant inclusively or exclusively.)

My main issue with this question is that I don't see how such a model would be constructed. For example, we can use a permutation model to create a single amorphous even (or odd) set, but how do we then conclude that every set is either even or odd? I don't know how to do this at all, so this motivates my second question:

Question 2: How does one build models of ZF+"there exists an amorphous set"+"every set is ---," in general? (Where --- denotes some arbitrary niceness property.)

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Every cardinal is finite or D-infinite. $\{\omega\}$ is certainly finite but not an element of $\omega$. :-) –  Asaf Karagila Jan 25 '12 at 21:05
    
Ouch, good point. Fixed. –  Noah S Jan 28 '12 at 0:39
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2 Answers

Much is known about the possibilities for amorphous sets, but not that much of it is known to me. You might begin by looking at John Truss's paper, "The structure of amorphous sets" (Ann. Pure Appl. Logic 73 (1995) 191-233; Math Reviews 96i:03047).

One easy fact that I do know, concerning your "inclusive or exclusive" remark on "even or odd": If a set is both even and odd, then it is Dedekind-infinite and therefore not amorphous. To see this, suppose you have a partition $P$ of $X$ into pairs, and you also have a partition $Q$ of $X-\{a\}$ into pairs. Then you can inductively define an injection $f$ of the natural numbers into $X$ by letting $f(0)$ be $a$, and thereafter letting $f(2n+1)$ be the element that is paired with $f(2n)$ in $P$, while $f(2n+2)$ is the element paired with $f(2n+1)$ in $Q$.

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First I'll remark that for the first question, note that the or cannot be exclusive since $\omega$ is both even and odd.

Now to take some definitions from [1] if $A$ is an amorphous set, let $U$ be a partition of $A$ into infinitely many parts. It follows that every $A\in U$ is finite, and all but finitely many have the same size. Let $n$ be that size (Truss calls this gauge of $U$).

We say that $A$ is strictly amorphous (also known as strongly amorphous) if there are no partitions with gauge $>1$; we say that $A$ is bounded amorphous if there is a finite bound on the possible gauges; and unbounded amorphous otherwise.

Some remarks about the first question:

Note that if $A$ is strictly amorphous then it is neither odd nor even. Also note that if $A$ is bounded amorphous (say by gauge $n$) then taking $U$ a partition of $A$ into parts of size $n$ and perhaps discarding a few elements gives us that $A/U$ is strictly amorphous:

Otherwise we could partition $A/U$ into pairs (or more) and the union of each part would give a partition of $A$ whose gauge is $>n$.

Therefore, to answer this question one first has to assure that no bounded amorphous set exists. If no bounded amorphous exists (which is consistent) then every amorphous set is either odd or even (depending on the partitions with gauge $2$).

There are a handful of examples to answer the second question in the paper.

I'll add a final remark that a bounded amorphous cannot have a group structure, but an unbounded amorphous can. In particular it is possible to have vector spaces which are amorphous sets (see my answers here and here).


Bibliography:

  1. J.K.Truss, The structure of amorphous sets, Annals of Pure and Applied Logic, 73 (1995), 191-233.
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