What are some "easy" violations of $\mathsf{SVC}$?

Question

By $\mathsf{SVC}$, I mean "small violations of choice", which is the statement $$(\exists S)(\forall X)(\exists f)``f\colon S\times\text{Ord}\to X\text{ is a surjection}".$$ Such an $S$ is called the seed. It is a standard, difficult to prove fact that the following are equivalent:

1. $V\vDash\mathsf{SVC}$;
2. $V\vDash(\exists A)(\forall X)(\exists f)``f\colon X\to A\times\text{Ord}\text{ is an injection}"$;
3. there is an model $W\subseteq V$ such that $W\vDash\mathsf{ZFC}$ and a symmetric system $\langle\mathbb{P},\mathscr{G},\mathscr{F}\rangle$ such that for some $W$-generic filter $G\subseteq\mathbb{P}$, $V=\mathsf{HS}^G_{\mathscr{F}}$; and
4. there is a notion of forcing $\mathbb{P}\in V$ such that $\mathbf{1}_{\mathbb{P}}\Vdash\mathsf{AC}$.

(I believe that all of these equivalences are collectively proved in [1] and [3], or are a relatively easy exercise.)

I am interested in looking at some consistency results concerning $\mathsf{ZF}+\lnot\mathsf{SVC}$, and would like examples of properties that are inconsistent with $\mathsf{SVC}$ for fairly clear reasons.

For example, in [2] Morris constructed a model of $\mathsf{ZF}$ in which for all ordinals $\alpha$, there is a set $A_\alpha$ that is a countable union of countable sets satisfying $\aleph_\alpha<\aleph^*(\mathscr{P}(A_\alpha))$. This is a violation of $\mathsf{SVC}$ because any extension in which choice holds, each $A_\alpha$ must be countable, and so $(2^{\aleph_0})^+$ must be larger than all of the ordinals.

References

[1] Blass, Andreas, Injectivity, projectivity, and the axiom of choice, Trans. Am. Math. Soc. 255, 31-59 (1979). MR0542870.

[2] Morris, Douglass Bert, Adding total indiscernibles to models of set theory, Ph.D. Thesis (1970). MR2620293.

[3] Usuba, Toshimichi, Choiceless Löwenheim-Skolem property and uniform definability of grounds, MR4378926.

Answer 1

There are two type of models of $\lnot\sf SVC$.

1. Take a symmetric system and vary its "base cardinal" and take a product thereof. This was done in many papers:

   • Asaf Karagila, Embedding orders into the cardinals with $\mathsf {DC}_{\kappa} $. Fundam. Math. 226 (2), 143-156 (2014) (ZBL1341.03068, arXiv:1212.4396).

   • A. Karagila, Fodor’s lemma can fail everywhere. Acta Math. Hung. 154 (1), 231-242 (2018) (ZBL1413.03012, arXiv:1610.03985).

   • G. P. Monro, Independence results concerning Dedekind-finite sets. J. Aust. Math. Soc., Ser. A 19, 35-46 (1975) (ZBL0298.02066).

   • Stanisław Roguski, A proper class of pairwise incomparable cardinals. Colloq. Math. 58 (2), 163-166 (1990) (ZBL0706.03038).

   • Marco Forti and Furio Honsell, The consistency of the axiom of universality for the ordering of cardinalities. J. Symb. Log. 50, 502-509 (1985) (ZBL0574.03038).

   And many many other papers of that vein.

2. Crazy models. These include models such as:

   • Asaf Karagila, The Bristol model: an abyss called a Cohen real. J. Math. Log. 18 (2), Article ID 1850008, 37 p. (2018) (ZBL06996242, arXiv:1704.06939).

   • Asaf Karagila, The Morris model. Proc. Am. Math. Soc. 148 (3), 1311-1323 (2020) (ZBL1477.03212, arXiv:1811.10977).

   • M. Gitik, All uncountable cardinals can be singular. Isr. J. Math. 35, 61-88 (1980) (ZBL0439.03036).

   And others as well (the second paper mentioned here is a variation on the original work of Morris mentioned in the question).

Some models straddle the line of these two things (for example the final model in https://arxiv.org/abs/1911.09285).

The distinction can be made with "is there an easily defined class forcing that forces back the axiom of choice" or not. Sometimes, such as the case in the Gitik & Morris models, there is no such extension (class generic or otherwise), and other times there is. One way of seeing this is by considering an inaccessible cardinal $\kappa$, and asking "If we did the construction below $\kappa$, can we force back the axiom of choice while preserving the inaccessibility of $\kappa$?" (the Bristol model straddles this line by making it very unclear if there is a class forcing that restores the axiom of choice or not, although I an inclined to believe that there is one).

Now. To your actual question about "properties" which I will interpret as "weak choice principles" (or its negation, "specific failures of choice"). It's kinda hard. To an extent, one can analyse the failure $\sf KWP$ in some of these models. For example, in Monro's model, which happens to be the same model as the one used by Roguski and by Forti–Honsell, $\sf KWP_1$ holds. It is not clear if $\sf BPI$ holds, though, and we don't have a good preservation theorem for $\sf BPI$ and I don't think anyone sat down to figure that one out.

In some of the other models, especially the ones made of "stitching local failures" (e.g. the revised Morris model from my paper above) we can sometimes bound the $\sf KWP$ failure on each local symmetric system and calculate the global failure. In some, such as the Bristol model (and the original Morris model, as far as I can tell, although this needs to be checked carefully), $\sf KWP$ fails entirely.

So, one way of producing a new result, perhaps, is to take a preservation theorem. For example the one from

• Asaf Karagila, Realizing realizability results with classical constructions. Bull. Symb. Log. 25 (4), 429-445 (2019) (ZBL07167381, arXiv:1905.08202).

Then using a stitching technique argue that a class-length stitching of local failures will also preserve this sort of failure. So, if successful, one can "somewhat straightforwardly" obtain a model where $\sf AC_{WO}$ holds but $\sf SVC$ fails.