Set Theory HT25, Classes
Flashcards
Basic definitions
@Define:
- A class $\mathbf X$
- What it means to say $a \in \mathbf X$
- What it means to say $\mathbf X$ and $\mathbf Y$ are equal
- What it means to say $\mathbf X \subseteq \mathbf Y$
- A class $\mathbf X$ is the collection of all sets satisfying a formula $\varphi$, i.e. $\lbrace x : \varphi(x)\rbrace$.
- $a \in \mathbf X$ if $\varphi(x)$
- $\mathbf X$ and $\mathbf Y$ are equal if $\forall x (\varphi(x) \leftrightarrow \psi(x))$, where $Y = \lbrace x : \psi(x)\rbrace$
- $\mathbf X \subseteq \mathbf Y$ if $\forall x(\phi(x) \rightarrow \psi(x))$
@Define a proper class.
A class which is not a set.
Class functions
@Define what is meant by a formula with parameters $\varphi(x, y)$ defining a class function $\mathbf F : \mathbf X \to \mathbf Y$.
Suppose:
- $\varphi(x, y)$ implies $x \in \mathbf X$ and $y \in \mathbf Y$, and
- for all $x \in \mathbf X$ there is a unique $y$ such that $\varphi(x, y)$ holds
and then $\mathbf F(x) = y$ means that $\varphi(x, y)$ holds.
Give a definition of the power set $\mathcal P(\cdot)$ as a class function.
$\mathcal P : \mathbf V \to \mathbf V$ is the class function defined by
\[\psi(x, y) := \forall w (w \in y \leftrightarrow w \subseteq x)\]@example~