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I cant really find a proper definition for this. If two sets are congruent, then what does that mean.

I heard that it can be defined in terms of isometries...

This is with respect to the banach tarski paradox/hausdorff paradox

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1 Answer

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Two sets $X$ and $Y$ are congruent it they can be each partititioned in $X=\coprod X_i$ and $Y=\coprod Y_i$ (finite partitions in the definition I know, I guess this may depend on the author), such that $X_i$ is isometric to $Y_i$.

I should probably add that "congruent" is kind of a buzzword in math, so it surely has a huuuge number of different definitions depending on the context, but I think this is what you're looking for.

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