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$\mathsf{SVC}(S)$ is the assertion that for all sets $X$ there is an ordinal $\eta$ and a surjection $f\colon\eta\times S\to X$. I would like to denote by $\mathsf{SVC}^\ast(S)$ the same assertion but with injections $f\colon X\to\eta\times S$. However, I seem to recall that $\mathsf{SVC}^\ast$ has already been used (perhaps even been canonised) for some other purpose. I haven't been able to find any references to it, but there are a lot of papers out there and asterisks are difficult to search for at the best of times.

Question: Does $\mathsf{SVC}^\ast$ have an established meaning in set theory?

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  • $\begingroup$ I think it'd be better to use a different symbol then * since that's usually to convert a notion regarding injective cardinalities into surjective. Maybe + since you're strengthening the notion? $\endgroup$ Commented May 14, 2024 at 21:47
  • $\begingroup$ @ElliotGlazer come to think of it that's a much better theory of why I didn't end up using $\mathsf{SVC}^\ast$... Also, thanks for the suggestion! $\mathsf{SVC^+}$ seems quite fitting. $\endgroup$ Commented May 20, 2024 at 8:50

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For what is worth, I find no hits on Math Reviews for $\mathsf{SVC}^*$ and all hits for $\mathsf{SVC}$ are for the statement that says (in your notation) that there is an $S$ such that $\mathsf{SVC}(S)$ holds.

(For the curious, $\mathsf{SVC}$, the axiom of "small violations of choice," was introduced by A. Blass in the late 1970s.)

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    $\begingroup$ Thank you for looking, I hadn't thought to use MR. $\endgroup$ Commented May 20, 2024 at 8:51

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