Well-Founded Induction

Theorem

Let $\struct {A, \RR}$ be a strictly well-founded relation.

Let $\RR^{-1} \sqbrk x$ denote the preimage of $x$ for each $x \in A$.

Let $B$ be a class such that $B \subseteq A$.

Suppose that:

$(1): \quad \forall x \in A: \paren {\RR^{-1} \sqbrk x \subseteq B \implies x \in B}$

Then:

$A = B$

That is, if a property passes from the preimage of $x$ to $x$, then this property is true for all $x \in A$.

Proof

Aiming for a contradiction, suppose $A \nsubseteq B$.

Then:

$A \setminus B \ne 0$

By Strictly Well-Founded Relation determines Strictly Minimal Elements, $A \setminus B$ must have some strictly minimal element under $\RR$.

Let $\map \complement B$ be the set complement of $B$.

Then:

\(\ds \exists x \in A \setminus B: \, \)

\(\ds \paren {A \setminus B} \cap \RR^{-1} \sqbrk x\)

\(=\)

\(\ds \O\)

\(\ds \leadsto \ \ \)

\(\ds \paren {A \cap \map \complement B} \cap \RR^{-1} \sqbrk x\)

\(=\)

\(\ds \O\)

Definition of Set Complement

\(\ds \leadsto \ \ \)

\(\ds A \cap \paren {\map \complement B \cap \RR^{-1} \sqbrk x }\)

\(=\)

\(\ds \O\)

Intersection is Associative

\(\ds \leadsto \ \ \)

\(\ds A \cap \paren {\RR^{-1} \sqbrk x \cap \map \complement B }\)

\(=\)

\(\ds \O\)

Intersection is Commutative

\(\ds \leadsto \ \ \)

\(\ds \paren {A \cap \RR^{-1} \sqbrk x } \cap \map \complement B\)

\(=\)

\(\ds \O\)

Intersection is Associative

So, by Intersection with Complement is Empty iff Subset:

$A \cap \RR^{-1} \sqbrk x \subseteq B$

Since $\RR^{-1} \sqbrk x \subseteq A$, by definition of a subset:

$\RR^{-1} \sqbrk x \subseteq B$

Thus, by hypothesis $(1)$:

$x \in B$

But this contradicts the fact that:

$x \in A \setminus B$

By Proof by Contradiction it follows that:

$A \setminus B = \O$

and so:

$A \subseteq B$

Therefore:

$A = B$

$\blacksquare$

Also see

Well-Ordered Induction, a weaker theorem that does not require the Axiom of Foundation.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.22$