nLab
induction

This entry is about induction in the sense of logic. For induction (functors) in representation theory see induced module, induced comodule, cohomological induction.


Contents

Idea

The principle of induction says that if a proposition nP(n)n \in \mathbb{N} \vdash P(n) that depends on the natural numbers has the property that

  1. it is true for 00 \in \mathbb{N};

  2. if it is true for nn \in \mathbb{N} then it is true for n+1n+1;

then it is true for every nn \in \mathbb{N}.

When formulated in one of the formalizations below, one finds that this principle is but the simplest special case of a very general notion of induction over inductive types. Other examples are induction over lists, trees, terms in a logic, and so on.

The dual notion is that of coinduction.

Formalization

By initial algebras over endofunctors

The induction principle my neatly be formalized in terms of the notion of initial algebras of an endofunctor.

In the category Set, the initial algebra for the endofunctor F:X1+XF: X \to 1 + X is 0,s:1+\langle 0, s \rangle : 1 + \mathbb{N} \to \mathbb{N}, where \mathbb{N} is the set of natural numbers, 00 is the smallest natural number, and ss is the successor operation.

In terms of this the principle of induction is equivalent to saying that there is no proper subalgebra of \mathbb{N}; that is, the only subalgebra is \mathbb{N} itself. This follows from the general property of initial objects that monomorphisms to them are isomorphisms.

References

  • Jiří Adámek, Stefan Milius, Lawrence Moss, Initial algebras and terminal coalgebras: a survey (pdf)

  • Bart Jacobs, Jan Rutten, A tutorial on (co)algebras and (co)induction, pdf EATCS Bulletin (1997); extended and updated version: An introduction to (co)algebras and (co)induction, In: D. Sangiorgi and J. Rutten (eds), Advanced topics in bisimulation and coinduction, p.38-99, 2011, pdf 62 pp.

Revised on November 21, 2013 09:11:34 by Zoran Škoda (161.53.130.104)