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mollwollfumble said:

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If you translate the funny symbols into words for us, I will respond with my profound thoughts on the question.

Or possibly just ask another question.

mollwollfumble said:

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Either you’re defining a new algebra, in which case the “0, 1, 2, ω” on the LHS of the equals sign are not numbers but symbols representing the sets, but then you also need to define the set your new algebra operates on, or the numbers on the LHS are the cardinalities of the sets, in which case your notation’s wrong. If the former, you need to supply more details of your algebra.

If the latter, |{}| = 0, |{0}| = 1, |{0,1}| = 2,
etc, to |{0,1,2,3,…}| == ω (=ℵ₀),
then using set-theory notation to link cardinalities is meaningless. If you’re just mixing notations to produce some pretty text, that’s fine, but again is meaningless. Rewriting your equations,

A = {}
B = {0}
C = {0,1}
etc.
Then |A| = 0, |B| = 1, |C| = 2, etc; B is clearly not in C, although it is a subset of C. Note that in set theory vertical bars around a set indicates the cardinality of the set.