# Question concerning Gödel's 'The consistency of the continuum hypothesis'

The work referred to is Kurt Gödel's 'The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory', Annals of Mathematical Studies number 3, Princeton University Press, 8th printing.

There is a primitive predicate $$\mathfrak{Cls}(A)$$ meaning '$$A$$ is a class'. My question is, isn't everything a class, making $$\mathfrak{Cls}$$ redundant? Compare -- some of the classes are sets and there is a primitive predicate $$\mathfrak{M}(A)$$ meaning '$$A$$ is a set', that I see the need for.

• $\mathfrak{Cls}(A)$ probably means "$A$ is a proper class". – Olivier Roche Dec 20 '19 at 12:40
• Haven't looked at the source, but it's worth noting that the collection of all classes satisfying a property might not be a class. Olivier's answer is almost surely the right one though. – Brevan Ellefsen Dec 20 '19 at 12:40
• @OlivierRoche $\mathfrak{Cls}(A)$ can't mean '$A$ is a proper class' because the first axiom says that every set is a $\mathfrak{Cls}$. – Justin Dec 20 '19 at 13:03
• Maybe it's there to rule out urelements? – Nagase Dec 20 '19 at 13:08
• Why are you reading Gödel's original paper? – Asaf Karagila Dec 20 '19 at 19:04

Without access I cannot be certain. Looking at what JSTOR has allowed, I am reminded of Goedel-Bernays or Von Neumann-Goedel-Bernays which uses a two-sorted logic so that one can speak of classes of sets where sets are classes that are elements of other classes.

Compare what you are reading with what is on Wikipedia as Von Neumann Bernays Goedel set theory.

Edit:

Found enough,

Hit free sample. You may get the same first pages.

Upper case variables are class variables, lower case variables are set variables.

Group A, Axiom 1

"Cls(x)"

translates to "all sets are classes"

Group A, Axiom 2

"If X is an element of Y, then M(X)'

translates to "Every class that is an element of a class is a set"

Group A, Axiom 3 is the extensionality of classes. Since only sets can be elements in this theory, its universal quantifier uses a lower case set variable.

• Why say all sets are classes? Everything is a class. There are no urelemente. – Justin Dec 24 '19 at 17:15
• The logic is not two-sorted. There are only class variable. $F(x)$ is just short for $(x)(\mathfrak{M}(x)\to F(x)$. – Justin Dec 24 '19 at 17:36
• What I'm reading is the monograph where von Neumann-Bernays-Gödel is first defined. Note also that the Wikipedia article is about a theory different from that in the monograph that I identify right at the start of my question. W. is about the theory in Mendelson's text Introduction to mathematical logic. – Justin Dec 29 '19 at 21:53