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I have read "Definable set".
But I sometimes see the word, "definable class".
For instance, von neumann universe $V = \{x | x=x\}$ is defined formally in meta theory.(ref. "What is the formal way to define “class” in ZFC?")
However if meta theory is $ZFC$, $V$ is not definable "class" but definable "set" because V is set in meta $ZFC$.

What is the meaning of "definable class"?

hina logic
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1 Answers1

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If you understand what is a definable set, think of the set theoretic universe as a set $V$ with some binary relation $E$, then a class is a subset of $V$ which is definable, with parameters, in the language of set theory.

The point is that we don't care about the meta-theory or the meta-universe. Those can be as strong as $\sf ZFC$ with additional axioms, or as weak as Primitive Recursive Arithmetic.

So we don't really refer to the meta-theory. We just talk about the class, as a collection. It is an object of the meta-universe, yes, either as a set there or as a formula with an assignment of parameters, or whatever you chose to work with.

The main point, and therein lies the strength of $\sf ZFC$ and its related cousins, is that given finitely many classes, we can sort of internalize them into the universe, and work with them almost as if they were sets, even though they are not. Which is exactly why we don't care so much about the meta-theory.

Asaf Karagila
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  • I understood. A definable set is a definable set by first-order language in $ZFC$ because we don't care about the meta theory. And so universe $V$ is a set or a formula, etc. in meta-theory, however, classes including $V$, $L$, and so on are treated like set in $ZFC$. Therefore them is definable "class" in first-order $ZFC$. Thank you for answering my question. – hina logic Mar 06 '19 at 13:15