Short question: Is there a standard term for a set $F$ such that there does not exist a surjection $F \twoheadrightarrow \omega$ (in the context of ZF)?
More detailed version: Consider the following four notions of “finiteness” in ZF, the third of which is the one I am asking about and will be arbitrarily named “P-finite” here:
“$F$ is finite” means any of the following equivalent statements:
there exists $n\in\omega$ and a bijection $n \xrightarrow{\sim} F$,
there exists a bijection $E \xrightarrow{\sim} F$ with $E\subseteq\omega$ and no bijection $\omega \xrightarrow{\sim} F$,
every nonempty subset of $\mathscr{P}(F)$ has a maximal element.
“$F$ is T-finite” means:
- every chain in $\mathscr{P}(F)$ has a maximal element.
“$F$ is P-finite” [nonstandard terminology which I'd like a standard term form] means any of the following equivalent statements:
$\mathscr{P}(F)$ is Noetherian under inclusion (i.e., any increasing sequence $A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots$ of subsets of $F$ is stationary),
$\mathscr{P}(F)$ is Artinian under inclusion (i.e., any decreasing sequence $A_0 \supseteq A_1 \supseteq A_2 \supseteq \cdots$ of subsets of $F$ is stationary),
there does not exist a surjection $F \twoheadrightarrow \omega$.
“$F$ is D-finite” (i.e., Dedekind-finite) means any of the following equivalent statements:
there is no bijection of $F$ with a proper subset of it,
there is no injection $\omega \hookrightarrow F$.
(I gave several equivalent conditions to emphasize the parallel between these four notions.)
We have finite $\Rightarrow$ T-finite $\Rightarrow$ P-finite $\Rightarrow$ D-finite, and none of the implications I just wrote is reversible. (To construct a permutation model with a P-finite set that is not T-finite, start with a set of atoms in bijection with $\mathbb{R}$ and use the group of permutations given by continuous increasing bijections $\mathbb{R} \xrightarrow{\sim} \mathbb{R}$ and the normal subgroup given by pointwise stabilizers of finite sets.)
Surely these four notions, and the implications and nonimplications I just mentioned must appear somewhere in the literature, as well as possibly others. My question is, what is the standard name for “P-finiteness”, and where are its properties, including what I just wrote, discussed in greater detail?
