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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.

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A couple of paragraphs above this property, in a version of Hungerford on Google Books here , there is this explanation:

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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.

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  • $\begingroup$ Is there any difference between "identify" and "define"? Because in ZFC the numbers are defined by sets, and this equation means the proper class $|I_{n}|=$a number. It is very weird. $\endgroup$ – Eric Sun Oct 13 '18 at 1:22
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    $\begingroup$ @EricYewenSun You'll have to accept that Hungerford's natural number $2$ (a certain proper class) is not the (von Neumann) natural number $2$ widely used in connection with ZFC (the set $\{\emptyset,\{\emptyset\}\}$). They also differ from Zermelo's $2$ (which I believe was $\{\{\emptyset\}\}$) and very probably from the $2$ you had in mind when you were a small child just learning to count (which was probably neither a set nor a proper class). $\endgroup$ – Andreas Blass Oct 13 '18 at 2:47
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    $\begingroup$ @Andreas: And also $2$ of the natural numbers as a model of PA ($SS0$) vs. $2$ of any ring, e.g. $\Bbb{Z,Q,R,C}$, which is $1+1$. $\endgroup$ – Asaf Karagila Oct 13 '18 at 7:39

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