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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3484/#Item_15" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about induction in the sense of <a class="existingWikiWord" href="/nlab/show/logic">logic</a>. For induction (functors) in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> see <em><a class="existingWikiWord" href="/nlab/show/induced+module">induced module</a></em>, <em><a class="existingWikiWord" href="/nlab/show/induced+comodule">induced comodule</a></em>, <em><a class="existingWikiWord" href="/nlab/show/cohomological+induction">cohomological induction</a></em>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="deduction_and_induction">Deduction and Induction</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/deductive+reasoning">deductive reasoning</a></strong>, <a class="existingWikiWord" href="/nlab/show/deduction">deduction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <p><a class="existingWikiWord" href="/nlab/show/hypothesis">hypothesis</a>/<a class="existingWikiWord" href="/nlab/show/context">context</a>/<a class="existingWikiWord" href="/nlab/show/antecedent">antecedent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_77a3b8c239221bcb6b2f5d78b1dd70489c654f80_1"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/conclusion">conclusion</a>/<a class="existingWikiWord" href="/nlab/show/consequence">consequence</a>/<a class="existingWikiWord" href="/nlab/show/succedent">succedent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+framework">logical framework</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deductive+system">deductive system</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/inductive+reasoning">inductive reasoning</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/induction">induction</a>, <a class="existingWikiWord" href="/nlab/show/recursion">recursion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></p> </li> </ul></div></div> <h4 id="induction">Induction</h4> <div class="hide"><div> <ul> <li><strong><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a></strong></li> <li><strong><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></strong></li> <li><strong><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></strong></li> </ul> <h2 id="sidebar_rules">Rules</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/induction">induction</a>, <a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></li> <li><a class="existingWikiWord" href="/nlab/show/recursion">recursion</a>, <a class="existingWikiWord" href="/nlab/show/corecursion">corecursion</a></li> </ul> <h2 id="sidebar_categorical_semantics">Categorical semantics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+algebra+of+an+endofunctor">initial algebra of an endofunctor</a>, <a class="existingWikiWord" href="/nlab/show/terminal+coalgebra+of+an+endofunctor">terminal coalgebra of an endofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+algebra+of+a+presentable+%E2%88%9E-monad">initial algebra of a presentable ∞-monad</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>, <a class="existingWikiWord" href="/nlab/show/natural+numbers+type">natural numbers type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/list">list</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-type">W-type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+type">circle type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+type">interval type</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#formalization'>Formalization</a></li> <ul> <li><a href='#by_initial_algebras_over_endofunctors'>By initial algebras over endofunctors</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <strong>principle of induction</strong> says that if a property of the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a></p> <ol> <li> <p>is <a class="existingWikiWord" href="/nlab/show/true">true</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{N}</annotation></semantics></math>;</p> </li> <li> <p>if it is true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> then it is true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>;</p> </li> </ol> <p>then it is true for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. The intuitive notion of a property is in practical mathematics identified with (belonging to) a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">S\subseteq\mathbb{N}</annotation></semantics></math>. Thus the mathematical induction says that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">0\in S</annotation></semantics></math> and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\forall n\in\mathbb{N}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>S</mi><mo>⇒</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">n\in S\implies n+1\in S</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">S = \mathbb{N}</annotation></semantics></math>. This is the way in the formal Dedekind-Peano arithmetics. Usually in logic one however uses a weaker form of induction which can be stated in a first order language (or in a type theory), namely where one limits to properties given by the predicates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(n)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dependent+type">depending</a> on natural numbers. The corresponding conclusion is the <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>⊢</mo><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n \in \mathbb{N} \vdash P(n)</annotation></semantics></math>. The latter version of the principle of induction is weaker because there are only <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\aleph_0</annotation></semantics></math> predicates but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">2^{\aleph_0}</annotation></semantics></math> subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>.</p> <p>When formulated in one of the formalizations below, one finds that the principle of induction for propositions depending on natural numbers is the simplest special case of a very general notion of induction over <em><a class="existingWikiWord" href="/nlab/show/inductive+types">inductive types</a></em>. Other examples are induction over <a class="existingWikiWord" href="/nlab/show/lists">lists</a>, <a class="existingWikiWord" href="/nlab/show/trees">trees</a>, terms in a <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, and so on.</p> <p>The <a class="existingWikiWord" href="/nlab/show/duality">dual</a> notion is that of <em><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></em>.</p> <h2 id="formalization">Formalization</h2> <h3 id="by_initial_algebras_over_endofunctors">By initial algebras over endofunctors</h3> <p>The induction principle may neatly be formalized in terms of the notion of <a class="existingWikiWord" href="/nlab/show/initial+algebras+of+an+endofunctor">initial algebras of an endofunctor</a>.</p> <p>In the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the <a class="existingWikiWord" href="/nlab/show/initial+algebra">initial algebra</a> <a class="existingWikiWord" href="/nlab/show/algebra+for+an+endofunctor">for the endofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn><mo>+</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F: X \to 1 + X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mn>0</mn><mo>,</mo><mi>s</mi><mo stretchy="false">⟩</mo><mo>:</mo><mn>1</mn><mo>+</mo><mi>ℕ</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\langle 0, s \rangle : 1 + \mathbb{N} \to \mathbb{N}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> is the set of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> is the smallest natural number, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/successor">successor</a> operation.</p> <p>In terms of this the <strong>principle of induction</strong> is equivalent to saying that there is no proper <a class="existingWikiWord" href="/nlab/show/subobject">subalgebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>; that is, the only subalgebra is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> itself. This follows from the general property of <a class="existingWikiWord" href="/nlab/show/initial+objects">initial objects</a> that <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> to them are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+induction">transfinite induction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deduction">deduction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inductive+definition">inductive definition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/recursion">recursion</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, Stefan Milius, Lawrence Moss, <em>Initial algebras and terminal coalgebras: a survey</em> (<a href="https://www.tu-braunschweig.de/Medien-DB/iti/survey_full.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bart+Jacobs">Bart Jacobs</a>, Jan Rutten, <em>A tutorial on (co)algebras and (co)induction</em>, <a href="http://www.cs.ru.nl/~bart/PAPERS/JR.pdf">pdf</a> EATCS Bulletin (1997); extended and updated version: <em>An introduction to (co)algebras and (co)induction</em>, In: D. Sangiorgi and J. Rutten (eds), Advanced topics in bisimulation and coinduction, p.38-99, 2011, <a href="http://www.cwi.nl/~janr/papers/files-of-papers/2011_Jacobs_Rutten_new.pdf">pdf</a> 62 pp.</p> </li> <li> <p>Ch. 3, Formal arithmetics, in: Elliott Mendelson, <em>Introduction to mathematical logic</em>, D. Van Nostrad 1964</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 17, 2022 at 22:58:05. 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