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3

After Benno van den Berg's proof, there are now two more proofs: First, building on his answer, Asaf removed the requirements for large cardinals in Asaf Karagila, Embedding Orders Into Cardinals With $DC_\kappa$, Fund. Math. 226 (2014), 143-156, doi:10.4064/fm226-2-4, arXiv:1212.4396. Second, as suggested by Mike Shulman, a suitable topos of $G$-sets ...


6

My "hunch" in the comments to the question appears to be correct! This model comes from Howard, Paul E.; Yorke, Mary F., Definitions of finite, Fundam. Math. 133, No. 3, 169-177 (1989). ZBL0704.03033. The paper has a few confusing typos and, in particular, the proof of Theorem~15 appears insufficient, so I'm sketching the argument in some detail, ...


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