# Epsilon-induction

In mathematics, ${\displaystyle \in }$-induction (epsilon-induction or set-induction) is a variant of transfinite induction.

Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction.

It can be used in set theory to prove that all sets satisfy a given property. This is a special case of well-founded induction.

## Statement

It states, all properties ${\displaystyle \Phi }$ , that if for every set ${\displaystyle x}$ , the truth of ${\displaystyle \Phi (x)}$  follows from the truth of ${\displaystyle \Phi }$  for all elements of ${\displaystyle x}$ , then this property ${\displaystyle \Phi }$  holds for all sets. In symbols:

${\displaystyle \forall x.{\Big (}\left(\forall y\in x.\Phi (y)\right)\rightarrow \Phi (x){\Big )}\ \rightarrow \ \forall z.\Phi (z)}$

Note that for the "bottom case" where ${\displaystyle x}$  denotes the empty set, the subexpression ${\displaystyle \forall y\in x.\Phi (y)}$  is vacuously true for all propositions.

### Comparison with natural number induction

The above can be compared with ${\displaystyle \omega }$ -induction over the natural numbers ${\displaystyle n\in \{0,1,2,\dots \}}$  for number properties ${\displaystyle \phi }$ . This may be expressed as

${\displaystyle {\Big (}\phi (0)\land \forall n.(\phi (n)\to \phi (n+1)){\Big )}\ \to \ \forall m.\phi (m)}$

or, using the symbol for the tautologically true statement, ${\displaystyle \top }$ ,

${\displaystyle {\Big (}(\top \to \phi (0))\land \forall n.(\phi (n)\to \phi (n+1)){\Big )}\ \to \ \forall m.\phi (m)}$

Fix a predicate ${\displaystyle Q}$  and define a new predicate equivalent to ${\displaystyle Q}$ , except with an argument offset by one and equal to ${\displaystyle \top }$  for ${\displaystyle 0}$ . With this we can also get a statement equivalent to induction for ${\displaystyle Q}$ , but without a conjunction. By abuse of notation, letting "${\displaystyle Q(-1)}$ " denote ${\displaystyle \top }$ , an instance of the ${\displaystyle \omega }$ -induction schema may thus be expressed as

${\displaystyle \forall n.{\Big (}Q(n-1)\to Q(n){\Big )}\ \to \ \forall m.Q(m)}$

This now mirrors an instance of the Set-Induction schema.

Conversely, Set-Induction may also be treated in a way that treats the bottom case explicitly.

#### Classical equivalents

With classical tautologies such as ${\displaystyle (A\to B)\iff (\neg A\lor B)}$  and ${\displaystyle (\neg A\lor B)\iff \neg (A\land \neg B)}$ , an instance of the ${\displaystyle \omega }$ -induction principle can be translated to the following statement:

${\displaystyle \exists n.{\Big (}Q(n-1)\land \neg Q(n){\Big )}\ \lor \ \forall m.Q(m)}$

This expresses that, for any property ${\displaystyle Q}$ , either there is any (first) number ${\displaystyle n}$  for which ${\displaystyle Q}$  does not hold, despite ${\displaystyle Q}$  holding for the preceding case, or - if there is no such failure case - ${\displaystyle Q}$  is true for all numbers.

Accordingly, in classical ZF, an instance of the set-induction can be translated to the following statement, clarifying what form of counter-example prevents a set-property ${\displaystyle P}$  to hold for all sets:

${\displaystyle \exists x.{\Big (}\left(\forall y\in x.P(y)\right)\,\land \,\neg P(x){\Big )}\ \lor \ \forall z\,P(z)}$

This expresses that, for any property ${\displaystyle P}$ , either there a set ${\displaystyle x}$  for which ${\displaystyle P}$  does not hold while ${\displaystyle P}$  being true for all elements of ${\displaystyle x}$ , or ${\displaystyle P}$  holds for all sets. For any property, if one can prove that ${\displaystyle \forall y\in x.P(y)}$  implies ${\displaystyle P(x)}$ , then the failure case is ruled out and the formula states that the disjunct ${\displaystyle \forall z.P(z)}$  must hold.

## Independence

In the context of the constructive set theory CZF, adopting the Axiom of regularity would imply the law of excluded middle and also set-induction. But then the resulting theory would be standard ZF. However, conversely, the set-induction implies neither of the two. In other words, with a constructive logic framework, set-induction as stated above is strictly weaker than regularity.