Geodesic
An example of two cardinals that are equivalent in the $n$-order logic and not equivalent in the $(n+1)$-order logic
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 1, pp. 35-44.

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It is proved that the property of two models to be equivalent in the $n$th order logic is definable in the $(n+1)$th order logic. Basing on this fact, there is given an (nonconstructive) “example” of two $n$-order equivalent cardinal numbers that are not $(n+1)$-order equivalent. 
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V. A. Bragin; E. I. Bunina. An example of two cardinals that are equivalent in the $n$-order logic and not equivalent in the $(n+1)$-order logic. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 1, pp. 35-44. http://geodesic.mathdoc.fr/item/FPM_2013_18_1_a2/

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