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Some questions and answers like this and this on Math Overflow suggest an ordering of large cardinals. I was wondering if there is any similar structure associated with $\text{ZF} + \neg \text{AC}$. In particular I am interested in the netherworld between finite and infinite.

For convenience augment the language of $\text{ZF}$ with a symbol $\omega$ for the least infinite ordinal. Let $\Psi_{i}(x)$ be the formula with one free variable that says there is a one-to-one map from $\omega$ into $x$. Let $\Psi_{f}(x)$ be the formula with one free variable that says for some element $n \in\omega$ there exists a one-to-one onto function from $n$ to $x$. The subscripts $i$ and $f$ are used to indicate versions of "infinite" and "finite".

Finally suppose we have a collection $\{ \Phi_{j} \colon j \in J \} $ of distinct formulas in one free variable. For each $j \in J$ suppose we have $$\text{ZF} + \text{AC} \vdash \forall x (\Phi_{j}(x) \implies \Psi_{i}(x)) $$

but it is consistent with $\text{ZF} + \neg \text{AC}$ for there to exist a set $x$ satisfying $\Phi_{j}(x) \& \neg \Psi_{f}(x) \& \neg \Psi_{i}(x)$.

Is it known that there is some kind of order for sets of formulas like $\{ \Phi_{j} \colon j \in J \} $ in the same way that there seems to be a nice order for large cardinal axioms? I don't know that it would be linear. That would be truly weird. What I am thinking about is something on the order of $$\text{ZF} \vdash \exists x \Phi_{0}(x) \implies \exists x \Phi_{1}(x).$$ I have picked one method of describing the space between finite and infinite. This is probably not the only way and may not be the best way. If there happens to be other descriptions for which actual results are known feel free to include those.

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Reading your question I am not really clear about what $\Phi$ is supposed to be? What sort of relationship are you trying to look for between infinite sets and the axiom of choice? The world of cardinalities in the absence of choice can become wild, very wild. – Asaf Karagila Aug 1 '11 at 11:44
I just wanted to comment that there is no actual need for $\omega$, since it is easily definable in $ZF$ alone. It is the least non-zero ordinal $\alpha$ which has an injection into a proper subset of itself. – Asaf Karagila Oct 4 '11 at 22:51
up vote 6 down vote accepted

It pretty much goes without saying that you should take a look at P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, volume 59 of Mathematical Surveys and Monographs. American Mathematical Society, 1998; it includes a great deal of relevant information. Updates are posted here. This thesis discusses a number of notions intermediate between finiteness and Dedekind finiteness. Both have numerous further references. The last six posts in this sci.math Usenet thread may also be of interest, though I suspect that everything in them is in Howard & Rubin.

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