Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are defined in terms of forcing. They seem interesting, at least to me, but I can't figure out even their most basic properties, even whether a nontrivial example of either exists.

Recall that a set $X$ is Dedekind-finite if there is no injection of $\omega$ into $X$. By results of Ellentuck and Sageev (http://logic-library.berkeley.edu/catalog/detail/739 and http://www.sciencedirect.com/science/article/pii/0003484381900176, respectively), it is consistent - relative to an inaccessible - that the Dedekind-finite cardinals form a nonstandard model $\mathcal{D}$ of true arithmetic. Specifically, Ellentuck showed that if the Dedekind-finite cardinalities are linearly ordered, then they form such a model, and Sageev showed the consistency of the former statement (plus $ZF+ACF$, where $ACF$ is the axiom of choice for finite sets).

Throughout the rest of this question, we live in such a model, $V$.

For the purposes of this question, say that a cut in a model of true arithmetic is subset of the model which is closed downwards and closed under successor. Since $\mathcal{D}\models TA$, there are no definable cuts in $\mathcal{D}$, but from an external perspective there may be many interesting cuts.

I'm interested in two particular classes of cuts, related to forcing, the first parametrized by Dedekind-finite cardinalities and the second parametrized by generic extensions of the universe:

• For $X$ and $Y$ Dedekind-finite cardinalities, write $$X\le_{DF} Y$$ if $X$ remains Dedekind-finite in every forcing extension in which $Y$ remains Dedekind-finite. For example, $X\le Y$ in the usual sense clearly implies $X\le_{DF} Y$, but equally clearly the converse is not true. Write $X<_{DF}Y$ if $X\le_{DF}Y$ but $Y\not\le_{DF}X$. Now define the cut under $Y$: $$C_u(Y):=\{X\in\mathcal{D}: X<_{DF}Y\}.$$ (Note that in any forcing extension witnessing $X_1<_{DF}Y$, if $X_0<X_1$ in $V$ then the inequality $X_0<X_1$ remains true in the forcing extension since otherwise $X_1$ would not remain Dedekind-finite.)

• The previous definition involves an existential quantifier over generic extensions. We can also consider the situation after a single generic extension: for $G$ generic over $V$, we define the cut after $G$: $$C_a(G):=\{X\in\mathcal{D}: V[G]\models\text{“X is Dedekind-finite''}\}.$$

Some immediate comments: $C_u$ and $C_a$ are always closed under $+$ and $\times$, since it is provable in ZF that the Dedekind-finite cardinalities are closed under $+$ and $\times$. However, it is not clear to me that $C_u$ and $C_a$ form models of true arithmetic, since they may not be the set of all Dedekind-finite cardinalities in the relevant forcing extensions.

So anyways, onto the question. What I want to ask is "Tell me everything you can about these cuts!" That, though, is a bit ridiculous. So here are two specific questions which I have spent some time on, and made exactly zero progress (partly because of the complexity of Sageev's construction, but also partly due to my own complete lack of expertise in both Choiceless set theory and nonstandard models of arithmetic):

Main Questions:

• Question 1.1: Are there always - or can there be - nontrivial (i.e. neither $\omega$ nor $\mathcal{D}$) cuts of the form $C_u$ and $C_a$?

• Question 1.2: Is every cut of the form $C_u$ also of the form $C_a$? Is every cut of the form $C_a$ also of the form $C_u$?

I mentioned above that Sageev's construction is (for me anyways) quite difficult; and in another question (Does Sageev's result need an inaccessible?), Asaf Karagila pointed out that Sageev's paper appears to be the end of the line in this sort of reasoning, and that any further work is likely to be extremely difficult. In light of this, here is why I think at least question 1.1 above is solvable: I suspect it is essentially a question about the combinatorics of forcing. In particular, a more technical question, but one of broader interest, which is relevant here:

Question 2: What is known about forcing in the context of Dedekind-finite cardinalities? For example, is there a straightforward description of $\le_{DF}$? Less ambitiously, is there a reasonable criterion for a forcing notion to add no new Dedekind-finite cardinalities?

(Two comments on this last question: first, over ZFC, no forcing adds new cardinalities, since every cardinality can be well-ordered assuming choice, and choice is preserved by forcing, and forcing does not change the ordinals; however, in our context, the first point is false and the second is inapplicable. Second, this question was previously touched upon in Asaf Karagila's question Forcing over models without the axiom of choice - there it was question 2 - but it was not addressed very much in the answers there.)

On the other hand, I do think that the following question, which to me is more interesting (if the answer to 1.1 is "yes," anyways :P), is likely to be impossibly hard (but I'll put it here anyways in case I'm wrong):

Question 3: What sort of arithmetic is true in cuts of the form $C_u$ and $C_a$? In particular, need they satisfy true arithmetic?

I can't access Ellentuck's paper, but it does not seem likely that an answer to this question falls immediately out of that work.

• Regarding future work, I hope to have good news in the near future. Nothing immediate for this question, but perhaps a step in the way of simplification of Sageev works. It's a bit hard to tell from here. – Asaf Karagila Jul 2 '14 at 16:47
• Cool - looking forward to hearing about it! – Noah Schweber Jul 2 '14 at 17:04

Here is an observation, which is both raising suspicion and hopes. I'm not sure which is which. (And this is really more of a comment, but it's just damn too long.)$\DeclareMathOperator{\Seq}{Seq}$
Let $\Seq(X)$ denote the set of injective finite sequences in $X^{<\omega}$. Then $X$ is Dedekind-finite if and only if $\Seq(X)$ is Dedekind-finite. In our universe all Dedekind-finite cardinals are comparable, therefore $X<|\Seq(X)|$. Note that $X$ and $\Seq(X)$ are not even in the same cut, since $\Seq(X)$ is larger than $X$ by orders of magnitude (it is closer to a power of $X$ than to multiplication).
But the cuts that you defined both rely on the fact that if $X$ remained Dedekind-finite then $\Seq(X)$ remained Dedekind-finite, and vice versa. Therefore the cuts $C_u$ and $C_a(G)$ are both closed not only under addition and multiplication, but under $\Seq$ as well.
On one side, this raises hopes that these cuts might exist and might satisfy enough axioms of arithmetics, even if not $\sf TA$ itself; but on the other side it shows that these cuts might be just too strong to be possible in the nontrivial case which is neither $\omega$ nor $\cal D$.
I'm not quite sure which case is true, though. And I'd be willing to guess that if one repeats Sageev's construction carefully enough with a $2$-inaccessible, then one might have such cuts after all. So their existence might be independent. It's really unclear at this moment, and any semi-definitive answer would certainly be surprising (if nothing else, then for not being surprising).