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No. The number of subsets of that set is strictly greater than the cardinality of that set, by Cantor's theorem. Moreover, it's consistent with ZFC that there are two sets which have different cardinality, yet have the same number of subsets.

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Q: Can we define the cardinal number as the number of subsets of that set?
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A fraction is a number, it is not a set. A number cannot have subsets, only a set can.

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