ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Veblen function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the Veblen functions are a hierarchy of <a href="Normal_function" title="Normal function">normal functions</a> (<a href="Continuous_function_(set_theory)" title="Continuous function (set theory)">continuous</a> <a href="Strictly_increasing_function" class="mw-redirect" title="Strictly increasing function">strictly increasing</a> <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> from <a href="Ordinal_number" title="Ordinal number">ordinals</a> to ordinals), introduced by <a href="Oswald_Veblen" title="Oswald Veblen">Oswald Veblen</a> in <a href="#CITEREFVeblen1908">Veblen (1908)</a>. If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> of φβ for β<α. These functions are all normal.

<div class="mw-heading mw-heading2"><h2 id="Veblen_hierarchy">Veblen hierarchy</h2></div>

In the special case when φ0(α)=ωα

this family of functions is known as the Veblen hierarchy.

The function φ1 is the same as the <a href="Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">ε function</a>: φ1(α)= εα.<a href="#cite_note-1">[1]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\beta \,,}">

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</math><img src="./1a9ff77845471100a86bd65e0483e5593646ddb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.952ex; height:2.509ex;" alt="{\displaystyle \alpha <\beta \,,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\alpha }(\varphi _{\beta }(\gamma ))=\varphi _{\beta }(\gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\alpha }(\varphi _{\beta }(\gamma ))=\varphi _{\beta }(\gamma )}</annotation>

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</math><img src="./7fce0f48fe032a558bf0892a17eec73b9f905ed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.244ex; height:3.009ex;" alt="{\displaystyle \varphi _{\alpha }(\varphi _{\beta }(\gamma ))=\varphi _{\beta }(\gamma )}" loading="lazy">.<a href="#cite_note-Rathjen90-2">[2]</a> From this and the fact that φβ is strictly increasing we get the ordering: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )}</annotation>

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</math><img src="./539a4c6da2909e2a98c5ce0ada36f607f116c14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.547ex; height:3.009ex;" alt="{\displaystyle \varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )}" loading="lazy"> if and only if either (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha <\gamma }</annotation>

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</math><img src="./0ef5c7601ba2d67261f1f13a6bff4d00581284bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:2.343ex;" alt="{\displaystyle \alpha <\gamma }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <\varphi _{\gamma }(\delta )}">

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<annotation encoding="application/x-tex">{\displaystyle \beta <\varphi _{\gamma }(\delta )}</annotation>

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</math><img src="./5b7a9a7a36dc278b20dd04b3a7996cc3d8a6641a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.933ex; height:3.009ex;" alt="{\displaystyle \beta <\varphi _{\gamma }(\delta )}" loading="lazy">) or (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >\gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha >\gamma }</annotation>

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</math><img src="./4593553ed3e2d55d7d6350ead1d4f5a703c15383.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:2.343ex;" alt="{\displaystyle \alpha >\gamma }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\alpha }(\beta )<\delta }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\alpha }(\beta )<\delta }</annotation>

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</math><img src="./133c0737b5ae941749cfb0e6d9d15d645e09edee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.093ex; height:2.843ex;" alt="{\displaystyle \varphi _{\alpha }(\beta )<\delta }" loading="lazy">).<a href="#cite_note-Rathjen90-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Fundamental_sequences_for_the_Veblen_hierarchy">Fundamental sequences for the Veblen hierarchy</h3></div>

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The fundamental sequence for an ordinal with <a href="Cofinality" title="Cofinality">cofinality</a> ω is a distinguished strictly increasing ω-sequence that has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image of n under the fundamental sequence for α will be indicated by α[n].

A variation of <a href="Ordinal_arithmetic#Cantor_normal_form" title="Ordinal arithmetic">Cantor normal form</a> used in connection with the Veblen hierarchy is: every nonzero ordinal number α can be uniquely written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\varphi _{\beta _{1}}(\gamma _{1})+\varphi _{\beta _{2}}(\gamma _{2})+\cdots +\varphi _{\beta _{k}}(\gamma _{k})}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =\varphi _{\beta _{1}}(\gamma _{1})+\varphi _{\beta _{2}}(\gamma _{2})+\cdots +\varphi _{\beta _{k}}(\gamma _{k})}</annotation>

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</math><img src="./94fd48c62cc2f4bb831ff9478a2167e13e2996a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.639ex; height:3.009ex;" alt="{\displaystyle \alpha =\varphi _{\beta _{1}}(\gamma _{1})+\varphi _{\beta _{2}}(\gamma _{2})+\cdots +\varphi _{\beta _{k}}(\gamma _{k})}" loading="lazy">, where k>0 is a natural number and each term after the first is less than or equal to the previous term, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta _{m}}(\gamma _{m})\geq \varphi _{\beta _{m+1}}(\gamma _{m+1})\,,}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta _{m}}(\gamma _{m})\geq \varphi _{\beta _{m+1}}(\gamma _{m+1})\,,}</annotation>

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</math><img src="./7348ec1def0489238aa9559a099a67a4ac4b077f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.352ex; height:3.176ex;" alt="{\displaystyle \varphi _{\beta _{m}}(\gamma _{m})\geq \varphi _{\beta _{m+1}}(\gamma _{m+1})\,,}" loading="lazy"> and each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{m}<\varphi _{\beta _{m}}(\gamma _{m}).}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{m}<\varphi _{\beta _{m}}(\gamma _{m}).}</annotation>

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</math><img src="./19164527fd03b5497d5ec3021a1035bf6ea8ae42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.332ex; height:3.009ex;" alt="{\displaystyle \gamma _{m}<\varphi _{\beta _{m}}(\gamma _{m}).}" loading="lazy"> If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha [n]=\varphi _{\beta _{1}}(\gamma _{1})+\cdots +\varphi _{\beta _{k-1}}(\gamma _{k-1})+(\varphi _{\beta _{k}}(\gamma _{k})[n])\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha [n]=\varphi _{\beta _{1}}(\gamma _{1})+\cdots +\varphi _{\beta _{k-1}}(\gamma _{k-1})+(\varphi _{\beta _{k}}(\gamma _{k})[n])\,.}</annotation>

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</math><img src="./49d6ed8ecbc9410e4c29d0f4113f3dbe5a55791a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:50.728ex; height:3.176ex;" alt="{\displaystyle \alpha [n]=\varphi _{\beta _{1}}(\gamma _{1})+\cdots +\varphi _{\beta _{k-1}}(\gamma _{k-1})+(\varphi _{\beta _{k}}(\gamma _{k})[n])\,.}" loading="lazy">

For any β, if γ is a limit with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma <\varphi _{\beta }(\gamma )\,,}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma <\varphi _{\beta }(\gamma )\,,}</annotation>

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</math><img src="./b3666427a4042cad2550f1c3be304bd6139e0c4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.161ex; height:3.009ex;" alt="{\displaystyle \gamma <\varphi _{\beta }(\gamma )\,,}" loading="lazy"> then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta }(\gamma )[n]=\varphi _{\beta }(\gamma [n])\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta }(\gamma )[n]=\varphi _{\beta }(\gamma [n])\,.}</annotation>

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</math><img src="./734c57b1b0de31d5928153669fdd13d0522f19ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.041ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta }(\gamma )[n]=\varphi _{\beta }(\gamma [n])\,.}" loading="lazy">

No such sequence can be provided for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{0}(0)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{0}(0)}</annotation>

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</math><img src="./d614f1bee85256fb819067beb00a38c6fd283b26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.546ex; height:2.843ex;" alt="{\displaystyle \varphi _{0}(0)}" loading="lazy"> = ω0 = 1 because it does not have cofinality ω.

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{0}(\gamma +1)=\omega ^{\gamma +1}=\omega ^{\gamma }\cdot \omega \,,}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{0}(\gamma +1)=\omega ^{\gamma +1}=\omega ^{\gamma }\cdot \omega \,,}</annotation>

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</math><img src="./93a4482db500027ac729ff032a4ebf706a72a8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.246ex; height:3.176ex;" alt="{\displaystyle \varphi _{0}(\gamma +1)=\omega ^{\gamma +1}=\omega ^{\gamma }\cdot \omega \,,}" loading="lazy"> we choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{0}(\gamma +1)[n]=\varphi _{0}(\gamma )\cdot n=\omega ^{\gamma }\cdot n\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{0}(\gamma +1)[n]=\varphi _{0}(\gamma )\cdot n=\omega ^{\gamma }\cdot n\,.}</annotation>

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</math><img src="./4bbdea2aabded917f73743ad650722eef635c734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.932ex; height:2.843ex;" alt="{\displaystyle \varphi _{0}(\gamma +1)[n]=\varphi _{0}(\gamma )\cdot n=\omega ^{\gamma }\cdot n\,.}" loading="lazy">

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta +1}(0)\,,}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta +1}(0)\,,}</annotation>

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</math><img src="./71459d745fb9e0e5b92efc9cd6cf9064fdd6aece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.8ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta +1}(0)\,,}" loading="lazy"> we use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta +1}(0)[0]=0}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta +1}(0)[0]=0}</annotation>

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</math><img src="./3b561894d6f1918c97b606d33420ba70c16f4e56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.484ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta +1}(0)[0]=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta +1}(0)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(0)[n])\,,}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta +1}(0)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(0)[n])\,,}</annotation>

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</math><img src="./10049804259e75a88d3a5022d205be959d6fe12d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.549ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta +1}(0)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(0)[n])\,,}" loading="lazy"> i.e. 0, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta }(0)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta }(0)}</annotation>

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</math><img src="./a8cb513db1117e488df261b2e261e681a43b2142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.666ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta }(0)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta }(\varphi _{\beta }(0))}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta }(\varphi _{\beta }(0))}</annotation>

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</math><img src="./6108a258a6a858047476c8ff3f3eadc68cda01c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.17ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta }(\varphi _{\beta }(0))}" loading="lazy">, etc..

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta +1}(\gamma +1)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta +1}(\gamma +1)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta +1}(\gamma +1)[0]=\varphi _{\beta +1}(\gamma )+1}</annotation>

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</math><img src="./df39d847f5d3ca5ae4e7f43f5dce29da0987fc77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.293ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta +1}(\gamma +1)[0]=\varphi _{\beta +1}(\gamma )+1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta +1}(\gamma +1)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(\gamma +1)[n])\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta +1}(\gamma +1)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(\gamma +1)[n])\,.}</annotation>

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</math><img src="./0b12e5c1468aff92a9f7d326fda3056a5da5a6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.754ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta +1}(\gamma +1)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(\gamma +1)[n])\,.}" loading="lazy">

Now suppose that β is a limit:

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <\varphi _{\beta }(0)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta <\varphi _{\beta }(0)}</annotation>

</semantics>

</math><img src="./b75514bbde70b7c3e0320107dfd117d855a87266.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.096ex; height:3.009ex;" alt="{\displaystyle \beta <\varphi _{\beta }(0)}" loading="lazy">, then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta }(0)[n]=\varphi _{\beta [n]}(0)\,.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta }(0)[n]=\varphi _{\beta [n]}(0)\,.}</annotation>

</semantics>

</math><img src="./bbf77d76626e9989b98184cc4bbe07788f324e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.054ex; height:3.176ex;" alt="{\displaystyle \varphi _{\beta }(0)[n]=\varphi _{\beta [n]}(0)\,.}" loading="lazy">

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta }(\gamma +1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta }(\gamma +1)}</annotation>

</semantics>

</math><img src="./01a6214b51b85dc37633ae48bce6887ab5804eaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.769ex; height:3.009ex;" alt="{\displaystyle \varphi _{\beta }(\gamma +1)}" loading="lazy">, use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\beta }(\gamma +1)[n]=\varphi _{\beta [n]}(\varphi _{\beta }(\gamma )+1)\,.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\beta }(\gamma +1)[n]=\varphi _{\beta [n]}(\varphi _{\beta }(\gamma )+1)\,.}</annotation>

</semantics>

</math><img src="./0bc7da2ff1ded694eac71f14e74535fec111ce21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.763ex; height:3.176ex;" alt="{\displaystyle \varphi _{\beta }(\gamma +1)[n]=\varphi _{\beta [n]}(\varphi _{\beta }(\gamma )+1)\,.}" loading="lazy">

Otherwise, the ordinal cannot be described in terms of smaller ordinals using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> and this scheme does not apply to it.

<div class="mw-heading mw-heading3"><h3 id="The_Γ_function">The Γ function</h3></div>

The function Γ enumerates the ordinals α such that φα(0) = α.

Γ0 is the <a href="Feferman%E2%80%93Sch%C3%BCtte_ordinal" title="Feferman–Schütte ordinal">Feferman–Schütte ordinal</a>, i.e. it is the smallest α such that φα(0) = α.

For Γ0, a fundamental sequence could be chosen to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}[0]=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}[0]=0}</annotation>

</semantics>

</math><img src="./293fa0078f63d33bbfb73cb3ad62b29f52c8ad50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.224ex; height:2.843ex;" alt="{\displaystyle \Gamma _{0}[0]=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}[n+1]=\varphi _{\Gamma _{0}[n]}(0)\,.}">

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}[n+1]=\varphi _{\Gamma _{0}[n]}(0)\,.}</annotation>

</semantics>

</math><img src="./d570d0812450b03287eeea1084c234176b358757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.815ex; height:3.176ex;" alt="{\displaystyle \Gamma _{0}[n+1]=\varphi _{\Gamma _{0}[n]}(0)\,.}" loading="lazy">

For Γβ+1, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\beta +1}[0]=\Gamma _{\beta }+1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\beta +1}[0]=\Gamma _{\beta }+1}</annotation>

</semantics>

</math><img src="./4200ac4aad2f91e7d41afd5ff2b436a925d97273.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.912ex; height:3.009ex;" alt="{\displaystyle \Gamma _{\beta +1}[0]=\Gamma _{\beta }+1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\beta +1}[n+1]=\varphi _{\Gamma _{\beta +1}[n]}(0)\,.}">

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\beta +1}[n+1]=\varphi _{\Gamma _{\beta +1}[n]}(0)\,.}</annotation>

</semantics>

</math><img src="./072af3f0ce062d062264e872b4fe7b05d7e6ef47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.838ex; height:3.343ex;" alt="{\displaystyle \Gamma _{\beta +1}[n+1]=\varphi _{\Gamma _{\beta +1}[n]}(0)\,.}" loading="lazy">

For Γβ where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta <\Gamma _{\beta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta <\Gamma _{\beta }}</annotation>

</semantics>

</math><img src="./c01290fdb085b3bd8c8ef7f3400b8b52bef2eeb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.057ex; height:2.843ex;" alt="{\displaystyle \beta <\Gamma _{\beta }}" loading="lazy"> is a limit, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\beta }[n]=\Gamma _{\beta [n]}\,.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\beta }[n]=\Gamma _{\beta [n]}\,.}</annotation>

</semantics>

</math><img src="./32bab2ea187370ea775b0ce2935de7fdd455d99c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.976ex; height:3.176ex;" alt="{\displaystyle \Gamma _{\beta }[n]=\Gamma _{\beta [n]}\,.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Finitely_many_variables">Finitely many variables</h3></div>

To build the Veblen function of a finite number of arguments (finitary Veblen function), let the binary function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\alpha ,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\alpha ,\gamma )}</annotation>

</semantics>

</math><img src="./85fd4a8b549ceeb8847d8f9faad68a2dad99771e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.113ex; height:2.843ex;" alt="{\displaystyle \varphi (\alpha ,\gamma )}" loading="lazy"> be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\alpha }(\gamma )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\alpha }(\gamma )}</annotation>

</semantics>

</math><img src="./623bc1627349e81890befd5ef2a894a2651187fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.876ex; height:2.843ex;" alt="{\displaystyle \varphi _{\alpha }(\gamma )}" loading="lazy"> as defined above.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> be an empty string or a string consisting of one or more comma-separated zeros <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,0,...,0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0,0,...,0}</annotation>

</semantics>

</math><img src="./b8f680c014fb02399f5a4c9406601282471caf08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0,0,...,0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> be an empty string or a string consisting of one or more comma-separated ordinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\alpha _{2},...,\alpha _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\alpha _{2},...,\alpha _{n}}</annotation>

</semantics>

</math><img src="./9350cdc6ec5aa82a9a421746f8c5eaa629e805e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.993ex; height:2.009ex;" alt="{\displaystyle \alpha _{1},\alpha _{2},...,\alpha _{n}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{1}>0}</annotation>

</semantics>

</math><img src="./f4c7a3f33ab88ed0446374557b3126581922bbba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.803ex; height:2.509ex;" alt="{\displaystyle \alpha _{1}>0}" loading="lazy">. The binary function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\beta ,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\beta ,\gamma )}</annotation>

</semantics>

</math><img src="./5726350febd02fe02fdd119fc08886dce67b49f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.958ex; height:2.843ex;" alt="{\displaystyle \varphi (\beta ,\gamma )}" loading="lazy"> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s,\beta ,z,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )}</annotation>

</semantics>

</math><img src="./2a78b130401af1b8958275c84ef65ae115a08e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.204ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )}" loading="lazy"> where both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> are empty strings.

The finitary Veblen functions are defined as follows:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\gamma )=\omega ^{\gamma }}</annotation>

</semantics>

</math><img src="./04656f09359e4363b912aa20e7d28f6217e8fc90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.261ex; height:2.843ex;" alt="{\displaystyle \varphi (\gamma )=\omega ^{\gamma }}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (z,s,\gamma )=\varphi (s,\gamma )}</annotation>

</semantics>

</math><img src="./a93e0a4b6e9b89a2c0225b216cf0dd971c00427a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.653ex; height:2.843ex;" alt="{\displaystyle \varphi (z,s,\gamma )=\varphi (s,\gamma )}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta >0}</annotation>

</semantics>

</math><img src="./4a87dc52878418173659e6d0ff8e77ab2897eac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >0}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s,\beta ,z,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )}</annotation>

</semantics>

</math><img src="./2a78b130401af1b8958275c84ef65ae115a08e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.204ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )}" loading="lazy"> denotes the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1+\gamma )}</annotation>

</semantics>

</math><img src="./cedd8dddd57644c4a58885c190517dc9fca925ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle (1+\gamma )}" loading="lazy">-th common fixed point of the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \varphi (s,\delta ,\xi ,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \varphi (s,\delta ,\xi ,z)}</annotation>

</semantics>

</math><img src="./383bbc7ad09ee607bc6002ccf6dfa7c4beab60ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.333ex; height:2.843ex;" alt="{\displaystyle \xi \mapsto \varphi (s,\delta ,\xi ,z)}" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta <\beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta <\beta }</annotation>

</semantics>

</math><img src="./502104299016ca59dd09876c1a4b139925a82fa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.479ex; height:2.676ex;" alt="{\displaystyle \delta <\beta }" loading="lazy"></li></ul>

For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,0,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (1,0,\gamma )}</annotation>

</semantics>

</math><img src="./fd68bd7d117e0f8034949c665d4eed5146deb134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \varphi (1,0,\gamma )}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1+\gamma )}</annotation>

</semantics>

</math><img src="./cedd8dddd57644c4a58885c190517dc9fca925ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle (1+\gamma )}" loading="lazy">-th fixed point of the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \varphi (\xi ,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \varphi (\xi ,0)}</annotation>

</semantics>

</math><img src="./f9976de73a339209e0473fe1a80b84befe7eb99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.2ex; height:2.843ex;" alt="{\displaystyle \xi \mapsto \varphi (\xi ,0)}" loading="lazy">, namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\gamma }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\gamma }}</annotation>

</semantics>

</math><img src="./4158379fb42d1aa5294ec5561eff8d9a0b7083de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.578ex; height:2.843ex;" alt="{\displaystyle \Gamma _{\gamma }}" loading="lazy">; then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,1,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (1,1,\gamma )}</annotation>

</semantics>

</math><img src="./4efe33c72bdd5679daa47758c11c3307462fbaa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \varphi (1,1,\gamma )}" loading="lazy"> enumerates the fixed points of that function, i.e., of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \Gamma _{\xi }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \Gamma _{\xi }}</annotation>

</semantics>

</math><img src="./f2b1832fb32a3e4b519779457718ca28d839d29f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.058ex; height:2.843ex;" alt="{\displaystyle \xi \mapsto \Gamma _{\xi }}" loading="lazy"> function; and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (2,0,\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (2,0,\gamma )}</annotation>

</semantics>

</math><img src="./8062b7ef0022079f0d9c2a8ee9d96570451322af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \varphi (2,0,\gamma )}" loading="lazy"> enumerates the fixed points of all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \varphi (1,\xi ,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \varphi (1,\xi ,0)}</annotation>

</semantics>

</math><img src="./38de5ad175a0a099888c8459b4c99c875b75d2dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.396ex; height:2.843ex;" alt="{\displaystyle \xi \mapsto \varphi (1,\xi ,0)}" loading="lazy">. Each instance of the generalized Veblen functions is continuous in the last nonzero variable (i.e., if one variable is made to vary and all later variables are kept constantly equal to zero).

The ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,0,0,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (1,0,0,0)}</annotation>

</semantics>

</math><img src="./6460ed5e3f816c6c69b2476e9667cb38f65385b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.081ex; height:2.843ex;" alt="{\displaystyle \varphi (1,0,0,0)}" loading="lazy"> is sometimes known as the <a href="Ackermann_ordinal" title="Ackermann ordinal">Ackermann ordinal</a>. The limit of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,0,...,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (1,0,...,0)}</annotation>

</semantics>

</math><img src="./d5d1191df748e25275b09080131da812cd202a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.02ex; height:2.843ex;" alt="{\displaystyle \varphi (1,0,...,0)}" loading="lazy"> where the number of zeroes ranges over ω, is sometimes known as the <a href="Small_Veblen_ordinal" title="Small Veblen ordinal">"small" Veblen ordinal</a>.

Every non-zero ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})}</annotation>

</semantics>

</math><img src="./ba95a4129ec6507a710e5f146e1049193b3af5ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.287ex; height:2.843ex;" alt="{\displaystyle \alpha =\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})}" loading="lazy">

where

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is a positive integer</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s_{1})\geq \varphi (s_{2})\geq \cdots \geq \varphi (s_{k})}</annotation>

</semantics>

</math><img src="./d6b5601c00dc24a282ee4caa9045fda59d7a1850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.475ex; height:2.843ex;" alt="{\displaystyle \varphi (s_{1})\geq \varphi (s_{2})\geq \cdots \geq \varphi (s_{k})}" loading="lazy"></li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{m}}</annotation>

</semantics>

</math><img src="./5afa6f6448663982b8719f5c6223e7212577a06a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.765ex; height:2.009ex;" alt="{\displaystyle s_{m}}" loading="lazy"> is a string consisting of one or more comma-separated ordinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{m,1},\alpha _{m,2},...,\alpha _{m,n_{m}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{m,1},\alpha _{m,2},...,\alpha _{m,n_{m}}}</annotation>

</semantics>

</math><img src="./86f196287cc784efd7ce2e02aaab2c714ef21c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.029ex; height:2.343ex;" alt="{\displaystyle \alpha _{m,1},\alpha _{m,2},...,\alpha _{m,n_{m}}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{m,1}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{m,1}>0}</annotation>

</semantics>

</math><img src="./e3e0da1ca4f9cd9b14979ecb5e5899fb5d478d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.703ex; height:2.843ex;" alt="{\displaystyle \alpha _{m,1}>0}" loading="lazy"> and each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{m,i}<\varphi (s_{m})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{m,i}<\varphi (s_{m})}</annotation>

</semantics>

</math><img src="./3a2f32cc7a65faa60822238acca5d4cdaab01d0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.381ex; height:3.009ex;" alt="{\displaystyle \alpha _{m,i}<\varphi (s_{m})}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Fundamental_sequences_for_limit_ordinals_of_finitary_Veblen_function">Fundamental sequences for limit ordinals of finitary Veblen function</h3></div>

For limit ordinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <SVO}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha <SVO}</annotation>

</semantics>

</math><img src="./16db548e9e60715697c5182bca3c22950970f167.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.646ex; height:2.176ex;" alt="{\displaystyle \alpha <SVO}" loading="lazy">, written in normal form for the finitary Veblen function:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k}))[n]=\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})[n]}</annotation>

</semantics>

</math><img src="./cf7bb4435f1d07f7c2d78825fba76d9f453ac338.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.687ex; height:2.843ex;" alt="{\displaystyle (\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k}))[n]=\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})[n]}" loading="lazy">,</li>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<mtext>is a successor ordinal</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<mtext>is a limit ordinal</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\gamma )[n]=\left\{{\begin{array}{lcr}n\quad {\text{if}}\quad \gamma =1\\\varphi (\gamma -1)\cdot n\quad {\text{if}}\quad \gamma \quad {\text{is a successor ordinal}}\\\varphi (\gamma [n])\quad {\text{if}}\quad \gamma \quad {\text{is a limit ordinal}}\\\end{array}}\right.}</annotation>

</semantics>

</math><img src="./2f65442fd313ee84a23560e45ae6a7f1ed73e3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:55.249ex; height:9.509ex;" alt="{\displaystyle \varphi (\gamma )[n]=\left\{{\begin{array}{lcr}n\quad {\text{if}}\quad \gamma =1\\\varphi (\gamma -1)\cdot n\quad {\text{if}}\quad \gamma \quad {\text{is a successor ordinal}}\\\varphi (\gamma [n])\quad {\text{if}}\quad \gamma \quad {\text{is a limit ordinal}}\\\end{array}}\right.}" loading="lazy">,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[0]=0}</annotation>

</semantics>

</math><img src="./39df6408191b318e2f58713cd966420f3daf219f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.921ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[0]=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)}</annotation>

</semantics>

</math><img src="./d81fcbb28e460382aeae115ff46e384a849afc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.831ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =0}</annotation>

</semantics>

</math><img src="./0a5e84cac32e896a80a89f8cd1917c2defcf4108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is a successor ordinal,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[0]=\varphi (s,\beta ,z,\gamma -1)+1}</annotation>

</semantics>

</math><img src="./36f992f175d9076f4db515f6c6d77f541a57c5a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.969ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[0]=\varphi (s,\beta ,z,\gamma -1)+1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)}</annotation>

</semantics>

</math><img src="./d81fcbb28e460382aeae115ff46e384a849afc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.831ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> are successor ordinals,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta ,z,\gamma [n])}</annotation>

</semantics>

</math><img src="./25a37ea0a8502f1e5c13f02e36fa96c28de1bea9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.884ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta ,z,\gamma [n])}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is a limit ordinal,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta [n],z,\gamma )}</annotation>

</semantics>

</math><img src="./54f5665257524ca79df7d053585ee4a8cf3adadd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.884ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta [n],z,\gamma )}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =0}</annotation>

</semantics>

</math><img src="./0a5e84cac32e896a80a89f8cd1917c2defcf4108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is a limit ordinal,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta [n],\varphi (s,\beta ,z,\gamma -1)+1,z)}</annotation>

</semantics>

</math><img src="./b5ae4daf681086eb27bdc1bf68f2acf5e83485f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.831ex; height:2.843ex;" alt="{\displaystyle \varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta [n],\varphi (s,\beta ,z,\gamma -1)+1,z)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is a successor ordinal and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is a limit ordinal.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Transfinitely_many_variables">Transfinitely many variables</h3></div>

More generally, Veblen showed that φ can be defined even for a transfinite sequence of ordinals αβ, provided that all but a finite number of them are zero. Notice that if such a sequence of ordinals is chosen from those less than an uncountable <a href="Regular_cardinal" title="Regular cardinal">regular cardinal</a> κ, then the sequence may be encoded as a single ordinal less than κκ (ordinal exponentiation). So one is defining a function φ from κκ into κ.

The definition can be given as follows: let α be a transfinite sequence of ordinals (i.e., an ordinal function with finite support) that ends in zero (i.e., such that α0=0), and let α[γ@0] denote the same function where the final 0 has been replaced by γ. Then γ↦φ(α[γ@0]) is defined as the function enumerating the common fixed points of all functions ξ↦φ(β) where β ranges over all sequences that are obtained by decreasing the smallest-indexed nonzero value of α and replacing some smaller-indexed value with the indeterminate ξ (i.e., β=α[ζ@ι0,ξ@ι] meaning that for the smallest index ι0 such that αι0 is nonzero the latter has been replaced by some value ζ<αι0 and that for some smaller index ι<ι0, the value αι=0 has been replaced with ξ).

For example, if α=(1@ω) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(1@ω) is the smallest fixed point of all the functions ξ↦φ(ξ,0,...,0) with finitely many final zeroes (it is also the limit of the φ(1,0,...,0) with finitely many zeroes, the small Veblen ordinal).

The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., that cannot be reached "from below" using the Veblen function of transfinitely many variables) is sometimes known as the <a href="Large_Veblen_ordinal" title="Large Veblen ordinal">"large" Veblen ordinal</a>, or "great" Veblen number.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Further_extensions">Further extensions</h3></div>

In <a href="#CITEREFMassmannKwon2023">Massmann & Kwon (2023)</a>, the Veblen function was extended further to a somewhat technical system known as dimensional Veblen. In this, one may take fixed points or row numbers, meaning expressions such as φ(1@(1,0)) are valid (representing the large Veblen ordinal), visualised as multi-dimensional arrays. It was proven that all ordinals below the <a href="Bachmann%E2%80%93Howard_ordinal" title="Bachmann–Howard ordinal">Bachmann–Howard ordinal</a> could be represented in this system, and that the representations for all ordinals below the <a href="Large_Veblen_ordinal" title="Large Veblen ordinal">large Veblen ordinal</a> were aesthetically the same as in the original system.

<div class="mw-heading mw-heading2"><h2 id="Values">Values</h2></div>

The function takes on several prominent values:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (1,0)=\varepsilon _{0}}</annotation>

</semantics>

</math><img src="./4a5a7cb8df60e72bb21734e890ed733bb2f0f502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.924ex; height:2.843ex;" alt="{\displaystyle \varphi (1,0)=\varepsilon _{0}}" loading="lazy"> is the <a href="Proof_theoretic_ordinal" class="mw-redirect" title="Proof theoretic ordinal">proof-theoretic ordinal</a> <a href="Gentzen's_consistency_proof" title="Gentzen's consistency proof">of</a> <a href="Peano_axioms" title="Peano axioms">Peano arithmetic</a> and the limit of what ordinals can be represented in terms of <a href="Cantor_normal_form" class="mw-redirect" title="Cantor normal form">Cantor normal form</a> and smaller ordinals.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\omega ,0)}</annotation>

</semantics>

</math><img src="./d432f3310d2848755f079c46de77e97acae80351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle \varphi (\omega ,0)}" loading="lazy">, a bound on the order types of the <a href="Path_ordering_(term_rewriting)" title="Path ordering (term rewriting)">recursive path orderings</a> with finitely many function symbols, and the smallest ordinal closed under <a href="Primitive_recursive_function" title="Primitive recursive function">primitive recursive</a> ordinal functions.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a></li>

<li>The <a href="Feferman%E2%80%93Sch%C3%BCtte_ordinal" title="Feferman–Schütte ordinal">Feferman–Schütte ordinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation>

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</math><img src="./44f27c3fa0660b68ef8fa747442140655cff65cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}" loading="lazy"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1,0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (1,0,0)}</annotation>

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</math><img src="./24de4e852aac1b36357313be3e88cd803fdfbdd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.885ex; height:2.843ex;" alt="{\displaystyle \varphi (1,0,0)}" loading="lazy">.<a href="#cite_note-6">[6]</a></li>

<li>The <a href="Small_Veblen_ordinal" title="Small Veblen ordinal">small Veblen ordinal</a> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi {\begin{pmatrix}1\\\omega \end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi {\begin{pmatrix}1\\\omega \end{pmatrix}}}</annotation>

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</math><img src="./72bc1f09788cf5102820d027ee13395048ee69d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.139ex; height:6.176ex;" alt="{\displaystyle \varphi {\begin{pmatrix}1\\\omega \end{pmatrix}}}" loading="lazy">.<a href="#cite_note-7">[7]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>Hilbert Levitz, <a rel="nofollow" class="external text" href="http://www.cs.fsu.edu/~levitz/ords.ps">Transfinite Ordinals and Their Notations: For The Uninitiated</a>, expository article (8 pages, in <a href="PostScript" title="PostScript">PostScript</a>)</li>

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</style><cite id="CITEREFPohlers1989" class="citation cs2">Pohlers, Wolfram (1989), <a rel="nofollow" class="external text" href="https://archive.org/details/prooftheoryintro0000pohl">Proof theory</a>, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-46825-7">10.1007/978-3-540-46825-7</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-51842-6</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1026933">1026933</a></cite></li>

<li><cite id="CITEREFSchütte1977" class="citation cs2">Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-07911-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0505313">0505313</a></cite></li>

<li><cite id="CITEREFTakeuti1987" class="citation cs2"><a href="Gaisi_Takeuti" title="Gaisi Takeuti">Takeuti, Gaisi</a> (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-87943-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0882549">0882549</a></cite></li>

<li><cite id="CITEREFSmorynski1982" class="citation cs2">Smorynski, C. (1982), "The varieties of arboreal experience", Math. Intelligencer, 4 (4): 182–189, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03023553">10.1007/BF03023553</a></cite> contains an informal description of the Veblen hierarchy.</li>

<li><cite id="CITEREFVeblen1908" class="citation cs2">Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society, 9 (3): 280–292, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1988605">10.2307/1988605</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1988605">1988605</a></cite></li>

<li><cite id="CITEREFMiller1976" class="citation cs2">Miller, Larry W. (1976), "Normal Functions and Constructive Ordinal Notations", The Journal of Symbolic Logic, 41 (2): 439–459, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2272243">10.2307/2272243</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2272243">2272243</a></cite></li>

<li><cite id="CITEREFMassmannKwon2023" class="citation cs2">Massmann, Jayde Sylvie; Kwon, Adrian Wang (October 20, 2023), <a rel="nofollow" class="external text" href="https://arxiv.org/abs/2310.12832">Extending the Veblen Function</a>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2310.12832">2310.12832</a></cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="Stephen_G._Simpson" class="mw-redirect" title="Stephen G. Simpson">Stephen G. Simpson</a>, Subsystems of Second-order Arithmetic (2009, p.387)

</li>

<li id="cite_note-Rathjen90-2">^ <a href="#cite_ref-Rathjen90_2-0">a</a> <a href="#cite_ref-Rathjen90_2-1">b</a> M. Rathjen, <a rel="nofollow" class="external text" href="https://www1.maths.leeds.ac.uk/~rathjen/Ord_Notation_Weakly_Mahlo.pdf">Ordinal notations based on a weakly Mahlo cardinal</a>, (1990, p.251). Accessed 16 August 2022.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> M. Rathjen, "<a rel="nofollow" class="external text" href="https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf">The Art of Ordinal Analysis</a>" (2006), appearing in Proceedings of the International Congress of Mathematicians 2006.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> N. Dershowitz, M. Okada, <a rel="nofollow" class="external text" href="https://www.cs.tau.ac.il/~nachumd/papers/ProofTheoretic.pdf">Proof Theoretic Techniques for Term Rewriting Theory</a> (1988). p.105

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFAvigad2001" class="citation journal cs1"><a href="Jeremy_Avigad" title="Jeremy Avigad">Avigad, Jeremy</a> (May 23, 2001). <a rel="nofollow" class="external text" href="https://www.andrew.cmu.edu/user/avigad/Papers/admissible.pdf">"An ordinal analysis of admissible set theory using recursion on ordinal notations"</a> (PDF). Journal of Mathematical Logic. 2: 91--112. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fs0219061302000126">10.1142/s0219061302000126</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> D. Madore, "<a rel="nofollow" class="external text" href="http://www.madore.org/~david/math/ordinal-zoo.pdf">A Zoo of Ordinals</a>" (2017). Accessed 02 November 2022.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFRanziStrahm2019" class="citation journal cs1">Ranzi, Florian; Strahm, Thomas (2019). <a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1007/s00153-019-00658-x.pdf">"A flexible type system for the small Veblen ordinal"</a> (PDF). Archive for Mathematical Logic. 58 (5–6): 711–751. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00153-019-00658-x">10.1007/s00153-019-00658-x</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:253675808">253675808</a>.</cite>

</li>

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