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Limitation of size: "For any class $C$, a set $x$ such that $x=C$ exists if and only if there is no bijection between $C$ and the class $V$ of all sets."

In Von Neumann–Bernays–Gödel set theory how could you state the axiom of limitation of size in first-order logic similarly to the axioms stated in first-order logic on the Wikipedia page?

I understand what a bijection is but I don't understand how you would refer to them in first-order logic (without just using a predicate).

Is there an answer using only predicates of equivalence and set/class membership?

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up vote 3 down vote accepted

First let us define some encodings:

  1. $\langle x,y\rangle = \{\{x\},\{x,y\}\}$ is an ordered pair.
  2. $R$ is a relation if and only if $\forall z\in R$ there are $x,y$ such that $z=\langle x,y\rangle$.
  3. $f$ is a function if and only if it is a relation and for every two pairs $\langle x,y\rangle\in f$ and $\langle a,b\rangle\in f$ if $x=a$ then $y=b$.
  4. $Dom(f)$ is the collection of all $x$ such that there is $y$ for which $\langle x,y\rangle\in f$, similarly we define $Rng(f)$.

Now we can say that $f$ is a function from class $X$ to the class $Y$ by specifying that $f$ is a function, its domain is exactly $X$ and its range is a subclass of $Y$.

To say that $f$ is also a bijection we say that every $y\in Y$ appears in the range of $f$ and that if $a\neq b$, and $\langle a,u\rangle,\langle b,v\rangle$ are both in $f$ then $u\neq v$ (if you think of $u=f(a)$ it is simply the condition $a\neq b\rightarrow f(a)\neq f(b)$).

Now to write that a class $C$ is a set if and only if there is no bijection from $C$ onto the universe it is simply to write the following formula:

For all $C$, $\exists D: C\in D$ if and only if for all $f$ which is a function, whose domain is $C$ there is some $x$ such that $x\notin Rng(f)$.

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