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<title>Hartogs number</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Hartogs number</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, specifically in <a href="Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>, a <b>Hartogs number</b> is an <a href="Ordinal_number" title="Ordinal number">ordinal number</a> associated with a set. In particular, if <i>X</i> is any <a href="Set_(mathematics)" title="Set (mathematics)">set</a>, then the Hartogs number of <i>X</i> is the least <a href="Ordinal_number" title="Ordinal number">ordinal</a> α such that there is no <a href="Injective_function" title="Injective function">injection</a> from α into <i>X</i>. If <i>X</i> can be <a href="Well-ordered" class="mw-redirect" title="Well-ordered">well-ordered</a> then the <a href="Cardinal_number" title="Cardinal number">cardinal number</a> of α is a minimal cardinal greater than that of <i>X</i>. If <i>X</i> cannot be well-ordered then there cannot be an injection from <i>X</i> to α. However, the cardinal number of α is still a minimal cardinal number (i.e. ordinal) <i>not less than or equal to</i> the cardinality of <i>X</i> (with the bijection definition of cardinality and the injective function order). (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not less than or equal to that of <i>X</i>.) The <a href="Map_(mathematics)" title="Map (mathematics)">map</a> taking <i>X</i> to α is sometimes called <b>Hartogs's function</b>. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets.

</p><p>The existence of the Hartogs number was proved by <a href="Friedrich_Hartogs" title="Friedrich Hartogs">Friedrich Hartogs</a> in 1915, using <a href="Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a> alone (that is, without using the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, or the later-introduced <a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">Replacement schema</a> of <a href="Zermelo-Fraenkel_set_theory" class="mw-redirect" title="Zermelo-Fraenkel set theory">Zermelo-Fraenkel set theory</a>).

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<div class="mw-heading mw-heading2"><h2 id="Hartogs's_theorem">Hartogs's theorem</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with the result in complex analysis, <a href="Hartogs's_theorem" class="mw-redirect" title="Hartogs's theorem">Hartogs's theorem</a>.</div>

<p>Hartogs's theorem states that for any set <i>X</i>, there exists an ordinal α such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |\not \leq |X|}">

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<annotation encoding="application/x-tex">{\displaystyle |\alpha |\not \leq |X|}</annotation>

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</math></span><img src="./ee51aab35ef304bca47000053bec394b5a61681c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.153ex; height:3.176ex;" alt="{\displaystyle |\alpha |\not \leq |X|}" loading="lazy"></span>; that is, such that there is no injection from α to <i>X</i>. As ordinals are well-ordered, this immediately implies the existence of a Hartogs number for any set <i>X</i>. Furthermore, the proof is constructive and yields the Hartogs number of <i>X</i>.

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

<p>See <a href="#CITEREFGoldrei1996">Goldrei 1996</a>.

</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\{\beta \in {\textrm {Ord}}\mid \exists i:\beta \hookrightarrow X\}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =\{\beta \in {\textrm {Ord}}\mid \exists i:\beta \hookrightarrow X\}}</annotation>

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</math></span><img src="./fad025513cd27d02b376dab011b07a49e42e5b2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.284ex; height:2.843ex;" alt="{\displaystyle \alpha =\{\beta \in {\textrm {Ord}}\mid \exists i:\beta \hookrightarrow X\}}" loading="lazy"></span> be the <a href="Class_(set_theory)" title="Class (set theory)">class</a> of all <a href="Ordinal_number" title="Ordinal number">ordinal numbers</a> <i>β</i> for which an <a href="Injective_function" title="Injective function">injective function</a> exists from <i>β</i> into <i>X</i>.

</p><p>First, we verify that <i>α</i> is a set.

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<ol><li><i>X</i> × <i>X</i> is a set, as can be seen in <a href="Axiom_of_power_set#Consequences" title="Axiom of power set">Axiom of power set</a>.</li>

<li>The <a href="Power_set" title="Power set">power set</a> of <i>X</i> × <i>X</i> is a set, by the axiom of power set.</li>

<li>The class <i>W</i> of all <a href="Reflexive_relation" title="Reflexive relation">reflexive</a> well-orderings of subsets of <i>X</i> is a definable subclass of the preceding set, so it is a set by the <a href="Axiom_schema_of_separation" class="mw-redirect" title="Axiom schema of separation">axiom schema of separation</a>.</li>

<li>The class of all <a href="Order_type" title="Order type">order types</a> of well-orderings in <i>W</i> is a set by the <a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">axiom schema of replacement</a>, as <style data-mw-deduplicate="TemplateStyles:r996643573">

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</style><div class="block-indent">(Domain(<i>w</i>), <i>w</i>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }">

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

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</math></span><img src="./a725ebc5ab8de11d7b71a8aa5a3706c2ea467885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }" loading="lazy"></span> (<i>β</i>, ≤)</div> can be described by a simple formula.</li></ol>

<p>But this last set is exactly <i>α</i>. Now, because a <a href="Transitive_set" title="Transitive set">transitive set</a> of ordinals is again an ordinal, <i>α</i> is an ordinal. Furthermore, there is no injection from <i>α</i> into <i>X</i>, because if there were, then we would get the contradiction that <i>α</i> ∈ <i>α</i>. And finally, <i>α</i> is the least such ordinal with no injection into <i>X</i>. This is true because, since <i>α</i> is an ordinal, for any <i>β</i> < <i>α</i>, <i>β</i> ∈ <i>α</i> so there is an injection from <i>β</i> into <i>X</i>.

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<div class="mw-heading mw-heading2"><h2 id="Historical_remark">Historical remark</h2></div>

<p>In 1915, Hartogs could use neither <a href="Ordinal_number#Von_Neumann_definition_of_ordinals" title="Ordinal number">von Neumann-ordinals</a> nor the <a href="Replacement_axiom" class="mw-redirect" title="Replacement axiom">replacement axiom</a>, and so his result is one of Zermelo set theory and looks rather different from the modern exposition above. Instead, he considered the set of isomorphism classes of well-ordered subsets of <i>X</i> and the relation in which the class of <i>A</i> precedes that of <i>B</i> if <i>A</i> is <a href="Order_isomorphism" title="Order isomorphism">isomorphic</a> with a proper initial segment of <i>B</i>. Hartogs showed this to be a well-ordering greater than any well-ordered subset of <i>X</i>. However, the main purpose of his contribution was to show that trichotomy for cardinal numbers implies the (then 11 year old) <a href="Well-ordering_theorem" title="Well-ordering theorem">well-ordering theorem</a> (and, hence, the axiom of choice).

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<ul><li><a href="Successor_cardinal" title="Successor cardinal">Successor cardinal</a></li>

<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li></ul>

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

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</style><cite id="CITEREFGoldrei1996" class="citation book cs1">Goldrei, Derek (1996). <i>Classic Set Theory</i>. <a href="Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>.</cite></li>

<li><cite id="CITEREFHartogs1915" class="citation journal cs1 cs1-prop-foreign-lang-source">Hartogs, Fritz (1915). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002266105">"Über das Problem der Wohlordnung"</a>. <i><a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i> (in German). <b>76</b> (4): <span class="nowrap">438–</span>443. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01458215">10.1007/BF01458215</a>. <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:45.0125.01">45.0125.01</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:121598654">121598654</a>.</cite></li>

<li><cite id="CITEREFJech,_Thomas2002" class="citation book cs1"><a href="Thomas_Jech" title="Thomas Jech">Jech, Thomas</a> (2002). <i>Set theory, third millennium edition (revised and expanded)</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-44085-2</bdi>.</cite></li>

<li><cite id="CITEREFCharles_Morgan" class="citation web cs1">Charles Morgan. <a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/~ucahcjm/ast/ast_notes_4.pdf">"Axiomatic set theory"</a> <span class="cs1-format">(PDF)</span>. <i>Course Notes</i>. University of Bristol<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-04-10</span></span>.</cite></li></ul>

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