Why are equivalence classes called so and not equivalence sets? I am kind of not able to find the difference between a class and a set. What properties that a set have that a class cannot have? It looks like classes are introduced to get away from Russel's paradox how does it?
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The Oxford English Dictionary s.v. equivalence quotes Kleene, Introduction to Metamathematics ($1952$), as follows:
This is the earliest citation that clearly refers to the usual modern sense of the term. It’s possible, then, that the English term is simply a calque of the German, and that the question should be why the German term uses Klasse rather than Menge.
The term "equivalence class" is a single term in the mathematical language. It does, however, relate to the term "class" from set theory.
Classes, in modern set theory, are collections which are defined by a formula (perhaps with parameters). An equivalence class is also defined by a formula with parameters. We have two parameters, the equivalence relation and the representative of the equivalence class.
In theories like $\sf ZFC$ every set is a class. Because every set is defined by a formula using itself as a parameter. It's a bit of cheating, but mathematically it is correct. In our case parameters are things we already know are sets, and if $A$ is a set then we can use it as a parameter for the formula $x\in A$.
The difference between proper classes, i.e. classes which are not sets, and sets has been discussed greatly on this site. In a nutshell proper classes are collection we can define (in the language of set theory) but we can prove that they do not form a set. One difference is that while we have a coherent way of assign "size" to a set, we don't have the ability to assign "size" to a proper class.