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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?

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3 Answers 3

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The term you might find in the literature is "weakly Dedekind finite", since a set that maps onto $\omega$ is weakly Dedekind infinite.

I'd expect that you'll call these "strongly Dedekind finite". Alas, these terms were coined before my arrival to this world. There is also "dually Dedekind finite", since we are defining Dedekind finitenesss with the dual notion of $\leq^*$, but that's not exactly the same thing either, since that refers to the case where "every surjection is a bijection", which is not equivalent, as it turns out, to "does not map onto $\omega$".

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I suggest the terminology "power Dedekind finite" by Andreas Blass in his paper Power-Dedekind Finiteness, and I use this terminology throughout all my papers. By Kuratowski's celebrated theorem, a set $F$ does not map onto $\omega$ if and only if the power set of $F$ is Dedekind finite. So this terminology does make sense. I do not like the terminology "weakly Dedekind finite", since it is in fact stronger than Dedekind finiteness. If we use "strongly Dedekind finite", then it contains a adverb "strongly" which is different from the adverb "weakly" used in its dual notion "weakly Dedekind infinite".

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    $\begingroup$ I think that's a good compromise. I'd even say "Power-finite" is probably a more compact term (and fits with P-finite). $\endgroup$
    – Asaf Karagila
    Dec 6, 2021 at 15:10
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My old paper

contains some (mildly interesting) constructions, and a (slightly more interesting) bibliography about notions of finiteness.

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