Reference request: definition of class

Question

I am looking for a reference on mathematically strict definition of class. For example, how do you construct the class of all ZFC-sets?

Answer 1

Within ZFC set theory, "class" is not a technical concept, so you won't find a definition of it.

It is common to use the word "class" to speak about all the sets that satisfy some particular property, as a collective, without implying that there is a set that has exactly those sets as elements. But -- at least from the point of view of the ZFC formalism -- this is an informal usage that doesn't correspond to anything in the theory.

When we're speaking about ZFC, seen from the outside, it is common to use the word "class" to speak about the collection of individuals of a model of ZFC that satisfy some particular formula. The formula can usually have parameters from the model, but this depends on exactly who is speaking and in which context. For example for the formula $x=x$ you get the class of all individuals in the model.

This usage is sometimes also imitated when we're speaking about ZFC purely as a syntactic proof system without any model in mind. In that case, however, a "class" is neither more nor less than a formula in the language of set theory, and speaking about it as a thing is just a suggestive shorthand without technical content.

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There are other set theories where classes do have a formal existence -- for example NBG set theory. But that won't give you a definition of "class" because in NBG, "class" is a fundamental, primitive concept -- A class is whatever behaves like the NBG axioms say classes behave.