/usr/share/doc/axiom-doc/hypertex/numcardinalnumbers.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Cardinal Numbers</div>
  <hr/>
The <a href="dbopcardinalnumber.xhtml">CardinalNumber</a> can be used for
values indicating the cardinality of sets, both finite and infinite. For
example, the <a href="dbopdimension.xhtml">dimension</a> operation in the
category <a href="dbopvectorspace.xhtml">VectorSpace</a> returns a cardinal
number.

The non-negative integers have a natural construction as cardinals
<pre>
0=#{ }, 1={0}, 2={0,1}, ..., n={i | 0 &#60;= i &#60; n}
</pre>
The fact that 0 acts as a zero for the multiplication of cardinals is
equivalent to the axiom of choice.

Cardinal numbers can be created by conversion from non-negative integers.
 <ul>
  <li>
   <input type="submit" id="p1" class="subbut" onclick="makeRequest('p1');"
     value="c0:=0::CardinalNumber" />
   <div id="ansp1"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p2" class="subbut" onclick="makeRequest('p2');"
     value="c1:=1::CardinalNumber" />
   <div id="ansp2"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p3" class="subbut" onclick="makeRequest('p3');"
     value="c2:=2::CardinalNumber" />
   <div id="ansp3"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p4" class="subbut" onclick="makeRequest('p4');"
     value="c3:=3::CardinalNumber" />
   <div id="ansp4"><div></div></div>
  </li>
 </ul>
The can also be obtained as the named cardinal Aleph(n)
 <ul>
  <li>
   <input type="submit" id="p5" class="subbut" onclick="makeRequest('p5');"
     value="A0:=Aleph 0" />
   <div id="ansp5"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p6" class="subbut" onclick="makeRequest('p6');"
     value="A1:=Aleph 1" />
   <div id="ansp6"><div></div></div>
  </li>
 </ul>
The <a href="dbopfiniteq.xhtml">finite?</a> operation tests whether a value
is a finite cardinal, that is, a non-negative integer.
 <ul>
  <li>
   <input type="submit" id="p7" class="subbut" 
     onclick="handleFree(['p3','p7']);"
     value="finite? c2" />
   <div id="ansp7"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p8" class="subbut" 
     onclick="handleFree(['p5','p8']);"
     value="finite? A0" />
   <div id="ansp8"><div></div></div>
  </li>
 </ul>
Similarly, the <a href="dbopcountableq.xhtml">countable?</a> operation
determines whether a value is a countable cardinal, that is, finite or
Aleph(0).
 <ul>
  <li>
   <input type="submit" id="p9" class="subbut" 
     onclick="handleFree(['p3','p9']);"
     value="countable? c2" />
   <div id="ansp9"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p10" class="subbut" 
     onclick="handleFree(['p5','p10']);"
     value="countable? A0" />
   <div id="ansp10"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p11" class="subbut" 
     onclick="handleFree(['p6','p11']);"
     value="countable? A1" />
   <div id="ansp11"><div></div></div>
  </li>
 </ul>
Arithmetic operations are defined on cardinal numbers as follows:
<table>
 <tr>
  <td>
   x+y = #(X+Y)
  </td>
  <td>
   cardinality of the disjoint union
  </td>
 </tr>
 <tr>
  <td>
   x-y = #(X-Y)
  </td>
  <td>
   cardinality of the relative complement
  </td>
 </tr>
 <tr>
  <td>
   x*y = #(X*Y)
  </td>
  <td>
   cardinality of the Cartesian product
  </td>
 </tr>
 <tr>
  <td>
   x+*y = #(X**Y)
  </td>
  <td>
   cardinality of the set of maps from Y to X
  </td>
 </tr>
</table>
Here are some arithmetic examples:
 <ul>
  <li>
   <input type="submit" id="p12" class="subbut" 
     onclick="handleFree(['p3','p6','p12']);"
     value="[c2+c2,c1+A1]" />
   <div id="ansp12"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p13" class="subbut" 
     onclick="handleFree(['p1','p2','p3','p5','p6','p13']);"
     value="[c0*c2,c1*c2,c2*c2,c0*A1,c1*A1,c2*A1,A0*A1]" />
   <div id="ansp13"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p14" class="subbut" 
     onclick="handleFree(['p1','p2','p3','p6','p14']);"
     value="[c2**c0,c2**c1,c2**c2,A1**c0,A1**c1,A1**c2]" />
   <div id="ansp14"><div></div></div>
  </li>
 </ul>
Subtraction is a partial operation; it is not defined when subtracting
a larger cardinal from a smaller one, nor when subtracting two equal
infinite cardinals.
 <ul>
  <li>
   <input type="submit" id="p15" class="subbut" 
     onclick="handleFree(['p2','p3','p4','p5','p6','p15']);"
     value="[c2-c1,c2-c2,c2-c3,A1-c2,A1-A0,A1-A1]" />
   <div id="ansp15"><div></div></div>
  </li>
 </ul>
The generalized continuum hypothesis asserts that
<pre>
 2**Aleph i = Aleph(i+1)
</pre>
and is independent of the axioms of set theory. (Goedel, The consistency
of the continuum hypothesis, Ann. Math. Studies, Princeton Univ. Press,
1940) The <a href="dbopcardinalnumber.xhtml">CardinalNumber</a> domain 
provides an operation to assert whether the hypothesis is to be assumed.
 <ul>
  <li>
   <input type="submit" id="p16" class="subbut" 
     onclick="makeRequest('p16');"
     value="generalizedContinuumHypothesisAssumed true" />
   <div id="ansp16"><div></div></div>
  </li>
 </ul>
When the generalized continuum hypothesis is assumed, exponentiation to
a transfinite power is allowed.
 <ul>
  <li>
   <input type="submit" id="p17" class="subbut" 
     onclick="handleFree(['p1','p2','p3','p5','p6','p17']);"
     value="[c0**A0,c1**A0,c2**A0,A0**A0,A0**A1,A1**A0,A1**A1]" />
   <div id="ansp17"><div></div></div>
  </li>
 </ul>
Three commonly encountered cardinal numbers are
<pre>
  a = #Z                 countable infinity
  c = #R                 the continuum
  f = #{g|g: [0,1]->R}
</pre>
In this domain, these values are obtained under the generalized continuum
hypothesis in this way:
 <ul>
  <li>
   <input type="submit" id="p18" class="subbut" 
     onclick="makeRequest('p18');"
     value="a:=Aleph 0" />
   <div id="ansp18"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p19" class="subbut" 
     onclick="handleFree(['p18','p19']);"
     value="c:=2**a" />
   <div id="ansp19"><div></div></div>
  </li>
  <li>
   <input type="submit" id="p20" class="subbut" 
     onclick="handleFree(['p18','p19','p20']);"
     value="f:=2**c" />
   <div id="ansp20"><div></div></div>
  </li>
 </ul>
 </body>
</html>
axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / numcardinalnumbers.xhtml
