Finite Class is Set
Theorem
Let $A$ be a finite class.
Then $A$ is a set.
Proof
Let it be assumed that all classes are subclasses of a basic universe $V$.
The proof proceeds by induction.
For all $n \in \N$, let $\map P n$ be the proposition:
- If $A$ is a finite class with $n$ elements, then $A$ is a set.
The Axiom of the Empty Set gives that the empty class $\O$ is a set.
From Empty Set is Unique, $\O$ is the only set (and hence class) with $0$ elements.
Thus $\map P 0$ is seen to hold.
Basis for the Induction
Let $A$ be a singleton class.
Thus by definition it has $1$ element, which we will call $x$.
Aiming for a contradiction, suppose $x$ is a class containing $1$ or more elements, one of which we may call $y$.
We have that $A$ is a subclass of $V$.
As $V$ is a basic universe, the Axiom of Transitivity holds.
Hence:
- $y \in x \land x \in A \implies y \in A$
But then $x \in A$ and $y \in A$ where $x \ne y$.
This contradicts our assertion that $A$ is a singleton class.
Hence:
- $x = \O$
It has been established that $\O$ is a set.
Hence $A$ is a class whose $1$ element is a set.
From Singleton Class of Set is Set, it follows that $A$ is a set.
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- If $A$ is a finite class with $k$ elements, then $A$ is a set.
from which it is to be shown that:
- If $A$ is a finite class with $k + 1$ elements, then $A$ is a set.
Induction Step
This is the induction step:
Let $A$ have $k + 1$ elements.
Let each of those elements be assigned a label:
- $a_1, a_2, \ldots, a_{k + 1}$
according to a bijection $\phi: A \to k$:
- $\map \phi {a_k} = k - 1$
This bijection $\phi$ is guaranteed to exist by definition of finite class.
Consider $A$ as the union of the classes:
- $\set {a_1, a_2, \ldots, a_k} \cup \set {a_{k + 1} }$
By the basis for the induction:
- $\set {a_{k + 1} }$ is a set.
By the induction hypothesis:
- $\set {a_1, a_2, \ldots, a_k}$ is a set.
By the Axiom of Unions:
- $\set {a_1, a_2, \ldots, a_k} \cup \set {a_{k + 1} }$ is a set.
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore, for all $\forall n \in \N$:
- If $A$ is a finite class with $n$ elements, then $A$ is a set.
That is, all finite classes are sets.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 6$ Finite Sets: Exercise $6.1 \ (1)$