We say that a model $M$ of $\mathsf{ZF}$ satisfies *Small Violations of Choice* ($\mathsf{SVC}$) if all (any) of the following apply:

- There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, and $M$ is a symmetric extension of $V$.
- There is a forcing $\mathbb{P}\in M$ such that $\mathbb{P}\mathrel{\Vdash}\mathsf{AC}$.
- There is $A\in M$ such that for all $X\in M$, there is an ordinal $\eta$ and a surjection $f\colon A\times\eta\to X$ in $M$.
- There is $A\in M$ such that for all $X\in M$, there is an ordinal $\eta$ and an injection $f\colon X\to A\times\eta$ in $M$.

The equivalence is true, standard, and very difficult. [1] introduces $\mathsf{SVC}$ and has some work towards proving their equivalence. For the purposes of this question, (1), (3), and (4) are not strictly required, but may be a helpful tool. What I am really interested in is "If (2) then can we force $\mathsf{AC}$ without identifying the cardinalities of distinct ordinals?".

I am particularly interested in condition (2), and in forcing $\mathsf{AC}$ in such a way that if $\lambda\neq\kappa\in M$ are well-ordered cardinals, $\mathbb{P}\mathrel{\Vdash}\check{\lambda}\neq\check{\kappa}$.

**Question 1.** Is this always possible? No! As Asaf Karagila has pointed out, we certainly require that all successors are regular. Any model of $\text{cf}(\omega_1)=\omega$ would not be able to be extended to a model of $\mathsf{AC}$ without collapsing $\omega_1$ to $\omega$.

**Question 2.** How much can we save? If $M\vDash\mathsf{DC}_{{<}\lambda}$, can we always keep cardinals at least the size of $\lambda$ distinct?

[1] *Blass, Andreas*, **Injectivity, projectivity, and the axiom of choice**, Trans. Am. Math. Soc. 255, 31-59 (1979). ZBL0426.03053.

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