Definition:Class (Class Theory)
Definition
A class is a collection of all sets such that a particular condition holds.
In class builder notation, this is written as:
- $\left\{{x : p \left({x}\right)}\right\}$
where $p \left({x}\right)$ is a statement containing $x$ as a free variable.
This is read:
- All $x$ such that $p \left({x}\right)$ holds.
Definition in ZF Set Theory
A class in $\textrm{ZF}$ is a formal vehicle capturing the intuitive notion of a class, namely a collection of all sets such that a particular condition $P$ holds.
In $\textrm{ZF}$, classes are written using class builder notation:
- $\left\{{x : P \left({x}\right)}\right\}$
Wouldn't you need the Class Comprehension Axiom for this?
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where $P \left({x}\right)$ is a statement containing $x$ as a free variable.
More formally, a class $\left\{ {x : P \left({x}\right)}\right\}$ serves to define the following definitional abbreviations involving the membership symbol:
| \(\displaystyle y \in \left\{ {x: P \left({x}\right)}\right\}\) | \(\quad \text{for} \quad\) | \(\displaystyle P \left({y}\right)\) | |||||||||||
| \(\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in y\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z \in y: \forall x: \left({x \in z \iff P \left({x}\right)}\right)\) | |||||||||||
| \(\displaystyle \left\{ {x: P \left({x}\right)}\right\} \in \left\{ {y: Q \left({y}\right)}\right\}\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z: \left({Q \left({z}\right) \land \forall x: \left({x \in z \iff P \left({x}\right)}\right)}\right)\) |
where:
- $x, y ,z$ are variables of $\textrm{ZF}$
- $P, Q$ are propositional functions.
Through these "rules", every statement involving $\left\{{x : P \left({x}\right) }\right\}$ can be reduced to a simpler statement involving only the basic language of set theory.
Proper Class
A proper class is a class which is not a set.
That is, $A$ is a proper class iff:
- $\neg \exists x: x = A$
where $x$ is a set.
A class which is not a proper class is a small class.
Also see
- Results about classes can be found here.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 4.2$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.1$: Sets
