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| <!-- 1. 非平凡初等嵌入 -->
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| <mtable columnalign="left">
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| <!-- 标题 -->
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| <mtr>
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| <mtd>
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| <mstyle mathvariant="bold" mathsize="1.2em">
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| <mtext>非平凡初等嵌入</mtext>
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| </mstyle>
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| </mtd>
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| </mtr>
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| <!-- 定义 -->
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| <mtr>
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| <mtd>
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| <mrow>
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| <mtext>设</mtext>
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| <mi>M</mi>
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| <mo>,</mo>
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| <mi>N</mi>
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| <mtext>为传递类且满足 ZF⁻；映射</mtext>
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| <mi>j</mi>
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| <mo>:</mo>
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| <mi>M</mi>
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| <mo>→</mo>
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| <mi>N</mi>
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| <mtext>称为初等嵌入当且仅当</mtext>
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| </mrow>
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| <mrow>
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| <mo>∀</mo>
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| <mi>φ</mi>
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| <mo>(</mo>
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| <msub><mi>x</mi><mn>1</mn></msub>
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| <mo>,</mo>
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| <mo>…</mo>
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| <mo>,</mo>
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| <msub><mi>x</mi><mi>n</mi></msub>
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| <mo>)</mo>
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| <mo>,</mo>
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| <mo>∀</mo>
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| <msub><mi>a</mi><mn>1</mn></msub>
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| <mo>,</mo>
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| <mo>…</mo>
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| <mo>,</mo>
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| <msub><mi>a</mi><mi>n</mi></msub>
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| <mo>∈</mo>
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| <mi>M</mi>
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| </mrow>
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| <mrow>
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| <mi>M</mi>
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| <mo>⊨</mo>
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| <mi>φ</mi>
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| <mo>[</mo>
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| <msub><mi>a</mi><mn>1</mn></msub>
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| <mo>,</mo>
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| <mo>…</mo>
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| <mo>,</mo>
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| <msub><mi>a</mi><mi>n</mi></msub>
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| <mo>]</mo>
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| <mo>⇔</mo>
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| <mi>N</mi>
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| <mo>⊨</mo>
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| <mi>φ</mi>
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| <mo>[</mo>
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| <mi>j</mi>
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| <mo>(</mo>
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| <msub><mi>a</mi><mn>1</mn></msub>
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| <mo>)</mo>
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| <mo>,</mo>
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| <mo>…</mo>
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| <mo>,</mo>
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| <mi>j</mi>
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| <mo>(</mo>
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| <msub><mi>a</mi><mi>n</mi></msub>
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| <mo>)</mo>
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| <mo>]</mo>
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| </mrow>
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| <mtext>；且称为</mtext>
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| <mstyle mathvariant="bold">
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| <mtext>非平凡</mtext>
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| </mstyle>
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| <mtext>当且仅当</mtext>
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| <mo>∃</mo>
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| <mi>x</mi>
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| <mo>∈</mo>
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| <mi>M</mi>
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| <mo>,</mo>
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| <mi>j</mi>
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| <mo>(</mo>
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| <mi>x</mi>
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| <mo>)</mo>
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| <mo>≠</mo>
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| <mi>x</mi>
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| <mo>.</mo>
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| </mtd>
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| </mtr>
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| <mtr><mtd><mspace height="0.6em"/></mtd></mtr>
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| <!-- 2. 临界点 -->
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| <mtr>
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| <mtd>
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| <mstyle mathvariant="bold" mathsize="1.2em">
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| <mtext>临界点</mtext>
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| </mstyle>
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| </mtd>
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| </mtr>
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| <mtr>
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| <mtd>
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| <mrow>
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| <mtext>对非平凡初等嵌入</mtext>
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| <mi>j</mi>
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| <mo>:</mo>
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| <mi>M</mi>
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| <mo>→</mo>
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| <mi>N</mi>
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| <mtext>，存在最小序数</mtext>
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| <mi>κ</mi>
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| <mtext>使得</mtext>
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| <mi>j</mi>
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| <mo>(</mo>
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| <mi>κ</mi>
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| <mo>)</mo>
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| <mo>≠</mo>
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| <mi>κ</mi>
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| <mo>，记</mtext>
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| <mtext>crit</mtext>
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| <mo>(</mo>
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| <mi>j</mi>
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| <mo>)</mo>
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| <mo>=</mo>
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| <mi>κ</mi>
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| <mo>.</mo>
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| </mrow>
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| </mtd>
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| </mtr>
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| <mtr><mtd><mspace height="0.6em"/></mtd></mtr>
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| <!-- 3. 共尾性 -->
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| <mtr>
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| <mtd>
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| <mstyle mathvariant="bold" mathsize="1.2em">
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| <mtext>共尾性</mtext>
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| </mstyle>
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| </mtd>
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| </mtr>
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| <mtr>
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| <mtd>
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| <mrow>
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| <mtext>嵌入</mtext>
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| <mi>j</mi>
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| <mo>:</mo>
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| <mi>M</mi>
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| <mo>→</mo>
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| <mi>N</mi>
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| <mtext>称为共尾，当且仅当</mtext>
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| <mo>∀</mo>
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| <mi>y</mi>
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| <mo>∈</mo>
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| <mi>N</mi>
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| <mo>,</mo>
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| <mo>∃</mo>
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| <mi>x</mi>
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| <mo>∈</mo>
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| <mi>M</mi>
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| <mo>,</mo>
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| <mi>y</mi>
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| <mo>∈</mo>
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| <mi>j</mi>
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| <mo>(</mo>
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| <mi>x</mi>
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| <mo>)</mo>
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| <mo>.</mo>
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| </mrow>
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| <mtext>若</mtext>
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| <mi>M</mi>
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| <mo>⊨</mo>
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| <mtext>ZF</mtext>
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| <mtext>且</mtext>
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| <mi>N</mi>
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| <mo>⊆</mo>
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| <mi>M</mi>
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| <mtext>，则任何初等嵌入都是共尾的</mtext>
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| <mo>.</mo>
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| </mtd>
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| </mtr>
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| <mtr><mtd><mspace height="0.6em"/></mtd></mtr>
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| <!-- 4. 一致性（Kunen 定理） -->
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| <mtr>
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| <mtd>
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| <mstyle mathvariant="bold" mathsize="1.2em">
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| <mtext>一致性</mtext>
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| </mstyle>
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| </mtd>
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| </mtr>
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| <mtr>
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| <mtd>
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| <mrow>
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| <mtext>在 ZFC 中不存在非平凡初等嵌入</mtext>
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| <mi>j</mi>
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| <mo>:</mo>
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| <mi>V</mi>
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| <mo>→</mo>
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| <mi>V</mi>
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| <mo>.</mo>
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| </mrow>
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| <mrow>
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| <mtext>更具体地（Kunen, 1971）：对任意序数</mtext>
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| <mi>λ</mi>
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| <mtext>，不存在非平凡初等嵌入</mtext>
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| <mi>j</mi>
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| <mo>:</mo>
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| <msub><mi>V</mi><mrow><mi>λ</mi><mo>+</mo><mn>2</mn></mrow></msub>
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| <mo>→</mo>
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| <msub><mi>V</mi><mrow><mi>λ</mi><mo>+</mo><mn>2</mn></mrow></msub>
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| <mtext>。</mtext>
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| </mrow>
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| </mtd>
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| </mtr>
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| </mtable>
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| <nowiki></math></nowiki>
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