In Hungerford's Algebra, he firstly defines that equipollent is a relation between two sets $A$, $B$ if there exists bijective $f:A\rightarrow B$, then proves
Equipollence is an equivalence relation on the class of all sets.
then
Definition 8.2: The cardinal number (or cardinality) of a set $A$, denoted $|A|$, is the equivalence class of $A$ under the equivalence relation of equipollence. $|A|$ is an infinite or finite cardinal according as $A$ is an infinite or finite set.
As he states in the next paragraph,
Cardinal numbers are frequently defined somewhat differently than we have done so that a cardinal number is in fact a set (instead of a proper class as in Definition 8.2)
I think I understand the definition and understand that the cardinal number is always a proper class.
But later the author asserts:
Property (iii) The cardinal number of a finite set is the number of elements in the set.
Here I can't understand it. By his definition, the cardinal number of a finite set should be an equivalence class by equipollence on the class of all sets, so how can it be a number? or it is how a number defined?
From the definition, I can only conclude that $|A|=|I_{n}|$ for some $n$, where $I_{n}=\{1,2,…,n\}$. So the property (iii) is true only when the "number" is actually a proper class? Hungerford does not clearly state what a "number" is, so I'm confused.
Best Answer
A couple of paragraphs above this property, in a version of Hungerford on Google Books here , there is this explanation:
In other words, Hungerford states that he is identifying the integer $n$ with the cardinal number that is the equivalence class under equipollency of sets with $n$ elements.