http://wiki.googology.top/index.php?action=history&feed=atom&title=Hakimi_Cardinal 2026-06-07T20:13:52Z 本wiki上该页面的版本历史 MediaWiki 1.43.1 http://wiki.googology.top/index.php?title=Hakimi_Cardinal&diff=3116&oldid=prev 2026-05-27T22:11:26Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="zh-Hans-CN"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2026年5月28日 (四) 06:11的版本</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">第5行：</td> <td colspan="2" class="diff-lineno">第5行：</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A cardinal k is a Hakimi cardinal if there exists a nonprincipal k-cofinally complete ultrafilter on k, alongside an elementary embedding j: L_k → L_2^k.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A cardinal k is a Hakimi cardinal if there exists a nonprincipal k-cofinally complete ultrafilter on k, alongside an elementary embedding j: L_k → L_2^k.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">&lt;nowiki&gt;**&lt;/nowiki&gt;</del>Lemma<del style="font-weight: bold; text-decoration: none;">** </del>(ZFC + A measurable cardinal exists): A Hakimi cardinal exists.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Lemma (ZFC + A measurable cardinal exists): A Hakimi cardinal exists.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">&lt;nowiki&gt;**&lt;/nowiki&gt;</del>Proof<del style="font-weight: bold; text-decoration: none;">**</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Proof</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We adopt this theorem: If a Ramsey cardinal exists, then for any uncountable cardinals k and λ, (Lk, ∈) is an elementary submodel of (Lλ, ∈).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We adopt this theorem: If a Ramsey cardinal exists, then for any uncountable cardinals k and λ, (Lk, ∈) is an elementary submodel of (Lλ, ∈).</div></td></tr> <!-- diff cache key my_wiki:diff:1.41:old-3115:rev-3116:php=table --> </table>

星汐镜Littlekk http://wiki.googology.top/index.php?title=Hakimi_Cardinal&diff=3115&oldid=prev 2026-05-27T22:10:56Z

<p>哈基米基数</p> <p><b>新页面</b></p><div>The Hakimi cardinal is a large cardinal notion proposed by Xuwang Zhihuan in 2026, and its relative consistency with ZFC was later proven.<br /> <br /> We define a k-cofinally complete filter over a cardinal k: A filter on k is k-cofinally complete if k has countable cofinality and the filter is σ-complete, or k has uncountable cofinality. For any a &lt; cf(k), the intersection of the family {Xb: b ∈ a} is contained in filter F whenever this family is a subset of F.<br /> <br /> A cardinal k is a Hakimi cardinal if there exists a nonprincipal k-cofinally complete ultrafilter on k, alongside an elementary embedding j: L_k → L_2^k.<br /> <br /> &lt;nowiki&gt;**&lt;/nowiki&gt;Lemma** (ZFC + A measurable cardinal exists): A Hakimi cardinal exists.<br /> <br /> &lt;nowiki&gt;**&lt;/nowiki&gt;Proof**<br /> <br /> We adopt this theorem: If a Ramsey cardinal exists, then for any uncountable cardinals k and λ, (Lk, ∈) is an elementary submodel of (Lλ, ∈).<br /> <br /> The existence of a measurable cardinal entails a Ramsey cardinal. Meanwhile, every measurable cardinal k has a nonprincipal k-cofinally complete ultrafilter. Since k and 2^k are both uncountable, it follows directly from the above theorem that k is a Hakimi cardinal. This completes the proof.<br /> <br /> It is conjectured that the consistency upper bound can be reduced to ZFC plus the existence of a Ramsey cardinal, but a full proof has not yet been completed.</div>

星汐镜Littlekk