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href="https://de.wikipedia.org/wiki/Limesordinalzahl" title="Limesordinalzahl – German" lang="de" hreflang="de" data-title="Limesordinalzahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ordinal_l%C3%ADmite" title="Ordinal límite – Spanish" lang="es" hreflang="es" data-title="Ordinal límite" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Lima_ordonombro" title="Lima ordonombro – Esperanto" lang="eo" hreflang="eo" data-title="Lima ordonombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ordinal_limite" title="Ordinal limite – French" lang="fr" hreflang="fr" data-title="Ordinal limite" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70894304 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%ED%95%9C_%EC%88%9C%EC%84%9C%EC%88%98" title="극한 순서수 – Korean" lang="ko" hreflang="ko" data-title="극한 순서수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ordinale_limite" title="Ordinale limite – Italian" lang="it" hreflang="it" data-title="Ordinale limite" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%B5%E9%99%90%E9%A0%86%E5%BA%8F%E6%95%B0" title="極限順序数 – Japanese" lang="ja" hreflang="ja" data-title="極限順序数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Graniczna_liczba_porz%C4%85dkowa" title="Graniczna liczba porządkowa – Polish" lang="pl" hreflang="pl" data-title="Graniczna liczba porządkowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D0%BB" title="Предельный ординал – Russian" lang="ru" hreflang="ru" data-title="Предельный ординал" data-language-autonym="Русский" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Omega-exp-omega-normal-dark_svg.svg/250px-Omega-exp-omega-normal-dark_svg.svg.png" decoding="async" width="250" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Omega-exp-omega-normal-dark_svg.svg/375px-Omega-exp-omega-normal-dark_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Omega-exp-omega-normal-dark_svg.svg/500px-Omega-exp-omega-normal-dark_svg.svg.png 2x" data-file-width="416" data-file-height="359" /></a><figcaption>Representation of the ordinal numbers up to ω<sup>ω</sup>. Each turn of the spiral represents one power of ω. Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω<sup>2</sup> </figcaption></figure> <p>In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, a <b>limit ordinal</b> is an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> that is neither zero nor a <a href="/wiki/Successor_ordinal" title="Successor ordinal">successor ordinal</a>. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, a successor ordinal, or a limit ordinal. </p><p>For example, the smallest limit ordinal is <a href="/wiki/%CE%A9_(ordinal_number)" class="mw-redirect" title="Ω (ordinal number)">ω</a>, the smallest ordinal greater than every <a href="/wiki/Natural_number" title="Natural number">natural number</a>. This is a limit ordinal because for any smaller ordinal (i.e., for any natural number) <i>n</i> we can find another natural number larger than it (e.g. <i>n</i>+1), but still less than ω. The next-smallest limit ordinal is ω+ω. This will be discussed further in the article. </p><p>Using the <a href="/wiki/Von_Neumann_definition_of_ordinals" class="mw-redirect" title="Von Neumann definition of ordinals">von Neumann definition of ordinals</a>, every ordinal is the <a href="/wiki/Well-ordered_set" class="mw-redirect" title="Well-ordered set">well-ordered set</a> of all smaller ordinals. The union of a nonempty set of ordinals that has no <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a> is then always a limit ordinal. Using <a href="/wiki/Von_Neumann_cardinal_assignment" title="Von Neumann cardinal assignment">von Neumann cardinal assignment</a>, every infinite <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> is also a limit ordinal. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Alternative_definitions">Alternative definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=1" title="Edit section: Alternative definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Various other ways to define limit ordinals are: </p> <ul><li>It is equal to the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> of all the ordinals below it, but is not zero. (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)</li> <li>It is not zero and has no maximum element.</li> <li>It can be written in the form ωα for α > 0. That is, in the <a href="/wiki/Ordinal_arithmetic#Cantor_normal_form" title="Ordinal arithmetic">Cantor normal form</a> there is no finite number as last term, and the ordinal is nonzero.</li> <li>It is a limit point of the class of ordinal numbers, with respect to the <a href="/wiki/Order_topology" title="Order topology">order topology</a>. (The other ordinals are <a href="/wiki/Isolated_point" title="Isolated point">isolated points</a>.)</li></ul> <p>Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor; some textbooks include 0 in the class of limit ordinals<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> while others exclude it.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because the <a href="/wiki/Class_(set_theory)" title="Class (set theory)">class</a> of ordinal numbers is <a href="/wiki/Well-order" title="Well-order">well-ordered</a>, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding <i>limit</i>), as it is the <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a> of the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>. Hence ω represents the <a href="/wiki/Order_type" title="Order type">order type</a> of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·<i>n</i> for any natural number <i>n</i>. Taking the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> (the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> operation on any <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of ordinals) of all the ω·n, we get ω·ω = ω<sup>2</sup>, which generalizes to ω<sup><i>n</i></sup> for any natural number <i>n</i>. This process can be further iterated as follows to produce: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{3},\omega ^{4},\ldots ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\ldots ,\varepsilon _{0}=\omega ^{\omega ^{\omega ^{~\cdot ^{~\cdot ^{~\cdot }}}}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mo>⋅</mo> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{3},\omega ^{4},\ldots ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\ldots ,\varepsilon _{0}=\omega ^{\omega ^{\omega ^{~\cdot ^{~\cdot ^{~\cdot }}}}},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f18137e87be9b22e4379f861f55693b83062cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.763ex; height:4.343ex;" alt="{\displaystyle \omega ^{3},\omega ^{4},\ldots ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\ldots ,\varepsilon _{0}=\omega ^{\omega ^{\omega ^{~\cdot ^{~\cdot ^{~\cdot }}}}},\ldots }" /></span></dd></dl> <p>In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> ordinals. However, there is no <a href="/wiki/Recursively_enumerable" class="mw-redirect" title="Recursively enumerable">recursively enumerable</a> scheme for <a href="/wiki/Ordinal_notation" title="Ordinal notation">systematically naming</a> all ordinals less than the <a href="/wiki/Church%E2%80%93Kleene_ordinal" class="mw-redirect" title="Church–Kleene ordinal">Church–Kleene ordinal</a>, which is a countable ordinal. </p><p>Beyond the countable, the <a href="/wiki/First_uncountable_ordinal" title="First uncountable ordinal">first uncountable ordinal</a> is usually denoted ω<sub>1</sub>. It is also a limit ordinal. </p><p>Continuing, one can obtain the following (all of which are now increasing in cardinality): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2},\omega _{3},\ldots ,\omega _{\omega },\omega _{\omega +1},\ldots ,\omega _{\omega _{\omega }},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{2},\omega _{3},\ldots ,\omega _{\omega },\omega _{\omega +1},\ldots ,\omega _{\omega _{\omega }},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbdab70d805e43be896d1025a933a27d5ff1506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.378ex; height:2.176ex;" alt="{\displaystyle \omega _{2},\omega _{3},\ldots ,\omega _{\omega },\omega _{\omega +1},\ldots ,\omega _{\omega _{\omega }},\ldots }" /></span></dd></dl> <p>In general, we always get a limit ordinal when taking the union of a nonempty set of ordinals that has no <a href="/wiki/Maximum" class="mw-redirect" title="Maximum">maximum</a> element. </p><p>The ordinals of the form ω²α, for α > 0, are limits of limits, etc. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classes of successor ordinals and limit ordinals (of various <a href="/wiki/Cofinality" title="Cofinality">cofinalities</a>) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a> or definitions by <a href="/wiki/Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a>. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> in the order topology and this is usually desirable. </p><p>If we use the <a href="/wiki/Von_Neumann_cardinal_assignment" title="Von Neumann cardinal assignment">von Neumann cardinal assignment</a>, every infinite <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> is also a limit ordinal (and this is a fitting observation, as <i>cardinal</i> derives from the Latin <i>cardo</i> meaning <i>hinge</i> or <i>turning point</i>): the proof of this fact is done by simply showing that every infinite successor ordinal is <a href="/wiki/Equinumerous" class="mw-redirect" title="Equinumerous">equinumerous</a> to a limit ordinal via the <a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hotel Infinity</a> argument. </p><p>Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level). </p> <div class="mw-heading mw-heading2"><h2 id="Indecomposable_ordinals">Indecomposable ordinals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=4" title="Edit section: Indecomposable ordinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Indecomposable_ordinal" class="mw-redirect" title="Indecomposable ordinal">Indecomposable ordinal</a></div> <p><b>Additively indecomposable</b> </p><p>A limit ordinal α is called additively indecomposable if it cannot be expressed as the sum of β < α ordinals less than α. These numbers are any ordinal of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b3346518f7829a4722251e2c474f371a891946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.62ex; height:2.676ex;" alt="{\displaystyle \omega ^{\beta }}" /></span> for β an ordinal. The smallest is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407f2fbc64e21bb432f397fdb816de0f5083d900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}" /></span>, the second is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2d6da217f4d6e937887bad1c90b371314830f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{1}}" /></span>, etc.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><b>Multiplicatively indecomposable</b> </p><p>A limit ordinal α is called multiplicatively indecomposable if it cannot be expressed as the product of β < α ordinals less than α. These numbers are any ordinal of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega ^{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{\omega ^{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d925e7ec5eb0fbec618c7931dcc65832fa742346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.629ex; height:3.009ex;" alt="{\displaystyle \omega ^{\omega ^{\beta }}}" /></span> for β an ordinal. The smallest is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4354a8b3e95ff6fbb81492f2774dfc11b4ef2a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.087ex; height:2.676ex;" alt="{\displaystyle \delta _{0}}" /></span>, the second is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797ad61e57939cd394800dfb549e859f144f7d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.087ex; height:2.676ex;" alt="{\displaystyle \delta _{1}}" /></span>, etc.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><b>Exponentially indecomposable and beyond</b> </p><p>The term "exponentially indecomposable" does not refer to ordinals not expressible as the exponential product <i>(?)</i> of β < α ordinals less than α, but rather the <a href="/wiki/Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">epsilon numbers</a>, "tetrationally indecomposable" refers to the zeta numbers, "pentationally indecomposable" refers to the eta numbers, etc.<sup id="cite_ref-:0_3-2" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">Ordinal arithmetic</a></li> <li><a href="/wiki/Limit_cardinal" title="Limit cardinal">Limit cardinal</a></li> <li><a href="/wiki/Fundamental_sequence_(ordinals)" class="mw-redirect" title="Fundamental sequence (ordinals)">Fundamental sequence (ordinals)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">for example, Thomas Jech, <i>Set Theory</i>. Third Millennium edition. Springer.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">for example, Kenneth Kunen, <i>Set Theory. An introduction to independence proofs</i>. North-Holland.</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://cantorsattic.info/Limit_ordinal#Types_of_Limits">"Limit ordinal - Cantor's Attic"</a>. <i>cantorsattic.info</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-08-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=cantorsattic.info&rft.atitle=Limit+ordinal+-+Cantor%27s+Attic&rft_id=http%3A%2F%2Fcantorsattic.info%2FLimit_ordinal%23Types_of_Limits&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALimit+ordinal" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Limit_ordinal&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor, G.</a>, (1897), <i>Beitrage zur Begrundung der transfiniten Mengenlehre. II</i> (tr.: Contributions to the Founding of the Theory of Transfinite Numbers II), Mathematische Annalen 49, 207-246 <a rel="nofollow" class="external text" href="https://archive.org/details/117770262">English translation</a>.</li> <li><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, J. H.</a> and <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, R. K.</a> "Cantor's Ordinal Numbers." In <i>The Book of Numbers</i>. New York: Springer-Verlag, pp. 266–267 and 274, 1996.</li> <li>Sierpiński, W. (1965). <i><a href="/wiki/Cardinal_and_Ordinal_Numbers" title="Cardinal and Ordinal Numbers">Cardinal and Ordinal Numbers</a></i> (2nd ed.). Warszawa: Państwowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.</li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Limit_ordinal&oldid=1274159075">https://en.wikipedia.org/w/index.php?title=Limit_ordinal&oldid=1274159075</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Ordinal_numbers" title="Category:Ordinal numbers">Ordinal numbers</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 5 February 2025, at 19:22<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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