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:
- $\set {x: \map p x}$
where $\map p x$ is a statement containing $x$ as a free variable.
This is read:
- All $x$ such that $\map p x$ 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:
- $\set {x : \map P x}$
Wouldn't you need the Class Comprehension Axiom for this?
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where $\map P x$ is a statement containing $x$ as a free variable.
More formally, a class $\set {x: \map P x}$ serves to define the following definitional abbreviations involving the membership symbol:
| \(\displaystyle y \in \set {x: \map P x}\) | \(\quad \text{for} \quad\) | \(\displaystyle \map P y\) | |||||||||||
| \(\displaystyle \set {x: \map P x} \in y\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z \in y: \forall x: \paren {x \in z \iff \map P x}\) | |||||||||||
| \(\displaystyle \set {x: \map P x} \in \set {y: \map Q y}\) | \(\quad \text{for} \quad\) | \(\displaystyle \exists z: \paren {\map Q z \land \forall x: \paren {x \in z \iff \map P x} }\) |
where:
- $x, y ,z$ are variables of $\textrm{ZF}$
- $P, Q$ are propositional functions.
Through these "rules", every statement involving $\set {x: \map P x}$ 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 if and only if:
- $\neg \exists x: x = A$
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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory
