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Cardinality and modulus share a symbol ($|a|$). Is cardinality just a special name for the modulus of a set?

Bernard
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    No. The cardinality of $A$ denotes the class of all the sets $B$ such that there is a bijection $A\to B$. In ZFC, the cardinality of $A$ may be identified with the least ordinal $\alpha$ such that there is a bijection $A\to \alpha$. –  May 20 '20 at 17:43
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    The symbol $|x|$ just means "the size of $x$ in some manner". Modulus, cardinality, determinant, measure, etc. all share the same symbol because they all somehow measure "size" (whatever that means). You shouldn't mix them up, though, as they will often be inconsistent with each other. – Brian Moehring May 20 '20 at 17:44
  • How does one define the modulus of a set? – gen-ℤ ready to perish May 20 '20 at 18:28
  • @gen-zreadytoperish At the intersection, we have objects like Dedekind cuts, so there are certainly some sets that have a well-defined modulus in the sense of analysis/geometry. – Brian Moehring May 20 '20 at 18:41

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The modulus operator $|\cdot|$ on $\mathbb{R}$ (or $\mathbb{C}$) acts on a different space than the "cardinality operator" $|\cdot|$, which acts on the space of sets. They are not the same operation, although they share some similarities - both give, in a certain sense, the "size" of the objects they act on. The modulus operator returns the magnitude of a complex number $\mathbb{C}$, while the cardinality of a set returns the number of elements a set contains. But the operations are certainly distinct for a multitude of reasons. Another reason is that the modulus operator is always a finite real value - for any $z\in\mathbb{C}$, $|z|\in\mathbb{R}$. The same cannot be said of the cardinality operator, which may take values in $\mathbb{N}$ or any of the "cardinal numbers" $\aleph_0,\aleph_1,\ldots$, and so forth. They simply do not even have the same domain and range.

csch2
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