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<p align="center"><b><font size="5">自 然 数</font></b></p>
<p align="center"> </p>
<p><b>后继集合:</b>设A为任意集合,A的后继集合记为A',定义为</p>
<p align="center">A'=A<img border="0" src="Image/bing.gif" width="14" height="14">{A}</p>
<p align="left">例如:A={a}, 则 A'=A<img border="0" src="Image/bing.gif" width="14" height="14">{A}={a}<img border="0" src="Image/bing.gif" width="14" height="14">{{a}}={a,{a}}</p>
<p align="left"> A={a,b}, 则 A'={a,b,{a,b}}</p>
<p align="left"> A=<img border="0" src="Image/kong.gif" width="11" height="11">,
则 A'=?</p>
<p align="left"> A={<img border="0" src="Image/kong.gif" width="11" height="11">},
则 A'=?</p>
<p align="left"> </p>
<p align="left"> </p>
<p align="left"><b>自然数N的定义(归纳法)</b></p>
<p align="left"><b> (1)(基础) <img border="0" src="Image/kong.gif" width="11" height="11"><img border="0" src="image/shuyu.gif" width="13" height="11">N.</b></p>
<p align="left"><b> (2)(归纳) 如果 n<img border="0" src="image/shuyu.gif" width="13" height="11">N,
则 n<img border="0" src="Image/bing.gif" width="14" height="14">{n}<img border="0" src="image/shuyu.gif" width="13" height="11">N
.</b></p>
<p align="left"><b> (3)(极小性) 如果 S<img src="image/baohan.gif" width="11" height="10">N
且满足(1)(2)规则, 则 S=N.</b></p>
<p align="left">自然数集合的元素为:</p>
<blockquote>
<p align="left"><img border="0" src="Image/kong.gif" width="11" height="11"></p>
<p align="left"><img border="0" src="Image/kong.gif" width="11" height="11"><b><img border="0" src="Image/bing.gif" width="14" height="14"></b>{<img border="0" src="Image/kong.gif" width="11" height="11">}
= {<img border="0" src="Image/kong.gif" width="11" height="11">}</p>
<p align="left">{<img border="0" src="Image/kong.gif" width="11" height="11">}<img border="0" src="Image/bing.gif" width="14" height="14">{{<img border="0" src="Image/kong.gif" width="11" height="11">}}
= {<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}</p>
<p align="left">{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}<img border="0" src="Image/bing.gif" width="14" height="14">{{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}}
= {<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}}</p>
<p align="left">{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}<img border="0" src="Image/bing.gif" width="14" height="14">{{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}}
= {<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}}}</p>
<p align="left">……</p>
</blockquote>
<p align="left">所以，自然数N=</p>
<p align="left"><b>{</b><img border="0" src="Image/kong.gif" width="11" height="11">、{<img border="0" src="Image/kong.gif" width="11" height="11">}、{<img border="0" src="Image/kong.gif" width="11" height="11"><img border="0" src="Image/bing.gif" width="14" height="14">{<img border="0" src="Image/kong.gif" width="11" height="11">}}、{<img border="0" src="Image/kong.gif" width="11" height="11"><img border="0" src="Image/bing.gif" width="14" height="14">{<img border="0" src="Image/kong.gif" width="11" height="11">}<img border="0" src="Image/bing.gif" width="14" height="14">{<img border="0" src="Image/kong.gif" width="11" height="11"><b><img border="0" src="Image/bing.gif" width="14" height="14"></b>{<img border="0" src="Image/kong.gif" width="11" height="11">}}}、{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">},{<img border="0" src="Image/kong.gif" width="11" height="11">,{<img border="0" src="Image/kong.gif" width="11" height="11">}}}}、……<b>}</b></p>
<p align="left"><b>为了方便起见</b>将自然数集合记为</p>
<p align="center">N={0，1，2，3，……}</p>
<p align="left"> </p>
<p align="left">下面这种自然数的定义是不对的：</p>
<p align="left"> (1)(基础) 0<img border="0" src="image/shuyu.gif" width="13" height="11">N.</p>
<p align="left"> (2)(归纳) 如果 n<img border="0" src="image/shuyu.gif" width="13" height="11">N,
则 n+1<img border="0" src="image/shuyu.gif" width="13" height="11">N .</p>
<p align="left"> (3)(极小性) 如果 S<img src="image/baohan.gif" width="11" height="10">N
且满足(1)(2)规则, 则 S=N.</p>
<p align="left"> 因为在上述定义中用到了“+”，而“+”是定义在N上的一种运算，这样变成了循环定义了。</p>
<p align="left"> </p>
<p style="line-height: 150%"> </p><p align="right"><b><a href="contentFrame-mulu.htm"><<back</a></b>
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