https://artofproblemsolving.com/wiki/index.php?title=Well_Ordering_Principle&feed=atom&action=historyWell Ordering Principle - Revision history2024-03-29T11:28:57ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=Well_Ordering_Principle&diff=173922&oldid=prevArr0w at 02:20, 13 May 20222022-05-13T02:20:34Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:20, 13 May 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; 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>The '''Well Ordering Principle''' states that every nonempty subset of the positive integers <math>\mathbb{Z}^{+}</math> contains a smallest element.  While this theorem is mostly brushed off as common sense, there is a bit of formalism required to actually prove <del class="diffchange diffchange-inline">the theorem </del>sufficiently.  We will do this here.   </div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The '''Well Ordering Principle''' states that every nonempty subset of the positive integers <math>\mathbb{Z}^{+}</math> contains a smallest element.  While this theorem is mostly brushed off as common sense, there is a bit of formalism required to actually prove <ins class="diffchange diffchange-inline">it </ins>sufficiently.  We will do this here.   </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>'''Definition''': A subset <math>A</math> of the real numbers is said to be inductive if it contains the number <math>1</math>, and if for every <math>x\in A</math>, the number <math>x+1\in A</math> as well.  Let <math>\mathcal{A}</math> be the collection of all the inductive subsets of <math>\mathbb{R}</math>.  Then the positive integers denoted <math>\mathbb{Z}^{+}</math> are defined by the equation</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>'''Definition''': A subset <math>A</math> of the real numbers is said to be inductive if it contains the number <math>1</math>, and if for every <math>x\in A</math>, the number <math>x+1\in A</math> as well.  Let <math>\mathcal{A}</math> be the collection of all the inductive subsets of <math>\mathbb{R}</math>.  Then the positive integers denoted <math>\mathbb{Z}^{+}</math> are defined by the equation</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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><center><math>\mathbb{Z}^{+}=\bigcap_{A\in \mathcal{A}}A</math></center></div></td><td class='diff-marker'>+</td><td style="color: #222; 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><center><math>\mathbb{Z}^{+}=\bigcap_{A\in \mathcal{A}}A</math><ins class="diffchange diffchange-inline">.</ins></center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>Using this definition, we can rephrase the principle of mathematical induction as follows: if <math>A</math> is an inductive set of the positive integers, then <math>A=\mathbb{Z}^{+}</math>.  We can now proceed with the proof.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>Using this definition, we can rephrase the principle of mathematical induction as follows: if <math>A</math> is an inductive set of the positive integers, then <math>A=\mathbb{Z}^{+}</math>.  We can now proceed with the proof.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>''Proof'': We first show that for any <math>n\in\mathbb{Z}^{+}</math>, every nonempty subset of <math>\{1,2,\ldots, n\}</math> has a smallest element.  Let <math>A</math> be the set of all positive integers <math>n</math> where this statement holds.  We see <math>A</math> contains <math>1</math>, since if <math>n=1</math> then the only subset of <math>\{1,2,\ldots, n\}</math> is <math>\{1\}</math> itself.  Then, supposing <math>A</math> contains <math>n</math> we show that it must contain <math>n+1</math>.  Let <math>C</math> be a nonempty subset of <math>\{1,2,\ldots, n+1\}</math>.  If <math>C=\{n+1\}</math> then <math>n+1</math> is its smallest element.  Otherwise, consider <math>C\cap \{1,2,\ldots,n\}</math>, which is nonempty.  Because <math>n\in A</math>, this set has a smallest element, which will be the smallest element of <math>C</math> also.  This means that <math>A</math> is inductive and <math>A=\mathbb{Z}^{+}</math>, so the statement is true for all <math>n\in\mathbb{Z}^{+}</math>.   </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>''Proof'': We first show that for any <math>n\in\mathbb{Z}^{+}</math>, every nonempty subset of <math>\{1,2,\ldots, n\}</math> has a smallest element.  Let <math>A</math> be the set of all positive integers <math>n</math> where this statement holds.  We see <math>A</math> contains <math>1</math>, since if <math>n=1</math> then the only subset of <math>\{1,2,\ldots, n\}</math> is <math>\{1\}</math> itself.  Then, supposing <math>A</math> contains <math>n</math> we show that it must contain <math>n+1</math>.  Let <math>C</math> be a nonempty subset of <math>\{1,2,\ldots, n+1\}</math>.  If <math>C=\{n+1\}</math> then <math>n+1</math> is its smallest element.  Otherwise, consider <math>C\cap \{1,2,\ldots,n\}</math>, which is nonempty.  Because <math>n\in A</math>, this set has a smallest element, which will be the smallest element of <math>C</math> also.  This means that <math>A</math> is inductive and <math>A=\mathbb{Z}^{+}</math>, so the statement is true for all <math>n\in\mathbb{Z}^{+}</math>.   </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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>Now suppose <math>D\in \mathbb{Z}^{+}</math> is nonempty.  By choosing some <math>n\in D</math>, the set <math>A\cap\{1,2\ldots, n\}</math> is also nonempty which means that <math>A</math> has a smallest element <math>k</math>.  This means that <math>k</math> is the smallest element of <math>D</math> too, which completes the proof<del class="diffchange diffchange-inline">.</del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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>Now suppose <math>D\in \mathbb{Z}^{+}</math> is nonempty.  By choosing some <math>n\in D</math>, the set <math>A\cap\{1,2\ldots, n\}</math> is also nonempty which means that <math>A</math> has a smallest element <math>k</math>.  This means that <math>k</math> is the smallest element of <math>D</math> too, which completes the proof <ins class="diffchange diffchange-inline"><math>\square</math></ins></div></td></tr>
</table>Arr0whttps://artofproblemsolving.com/wiki/index.php?title=Well_Ordering_Principle&diff=173921&oldid=prevArr0w: This article had a horrendous proof that was really informal. I added significant rigor and formalism to this topic.2022-05-13T02:19:43Z<p>This article had a horrendous proof that was really informal. I added significant rigor and formalism to this topic.</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:19, 13 May 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; 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>The '''Well Ordering Principle''' states that every nonempty <del class="diffchange diffchange-inline">set </del>of positive integers contains a smallest <del class="diffchange diffchange-inline">member</del>. <del class="diffchange diffchange-inline">The proof of </del>this is <del class="diffchange diffchange-inline">simply </del>common sense, <del class="diffchange diffchange-inline">but we can construct a formal proof by contradiction. Assume </del>there is <del class="diffchange diffchange-inline">no smallest element. Then for each element in </del>the <del class="diffchange diffchange-inline">set, there exists a smaller element, so if we take this smaller element, there must a different smaller element, and so on</del>. <del class="diffchange diffchange-inline">Since the set is finite, we cannot continue like </del>this <del class="diffchange diffchange-inline">infinitely many times, contradiction</del>.</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The '''Well Ordering Principle''' states that every nonempty <ins class="diffchange diffchange-inline">subset </ins>of <ins class="diffchange diffchange-inline">the </ins>positive integers <ins class="diffchange diffchange-inline"><math>\mathbb{Z}^{+}</math> </ins>contains a smallest <ins class="diffchange diffchange-inline">element</ins>. <ins class="diffchange diffchange-inline"> While </ins>this <ins class="diffchange diffchange-inline">theorem </ins>is <ins class="diffchange diffchange-inline">mostly brushed off as </ins>common sense, there is <ins class="diffchange diffchange-inline">a bit of formalism required to actually prove </ins>the <ins class="diffchange diffchange-inline">theorem sufficiently</ins>. <ins class="diffchange diffchange-inline"> We will do </ins>this <ins class="diffchange diffchange-inline">here</ins>. <ins class="diffchange diffchange-inline">  </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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 class="diffchange diffchange-inline">stub</del>}}</div></td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">'''Definition''': A subset <math>A</math> of the real numbers is said to be inductive if it contains the number <math>1</math>, and if for every <math>x\in A</math>, the number <math>x+1\in A</math> as well.  Let <math>\mathcal{A}</math> be the collection of all the inductive subsets of <math>\mathbb{R}</math>.  Then the positive integers denoted <math>\mathbb{Z}^</ins>{<ins class="diffchange diffchange-inline">+}</math> are defined by the equation</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; 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 class="diffchange diffchange-inline">[[Category</del>:<del class="diffchange diffchange-inline">Axioms]]</del></div></td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline"><center><math>\mathbb</ins>{<ins class="diffchange diffchange-inline">Z</ins>}<ins class="diffchange diffchange-inline">^{+</ins>}<ins class="diffchange diffchange-inline">=\bigcap_{A\in \mathcal{A}}A</math></center></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">Using this definition, we can rephrase the principle of mathematical induction as follows</ins>: <ins class="diffchange diffchange-inline">if <math>A</math> is an inductive set of the positive integers, then <math>A=\mathbb{Z}^{+}</math>.  We can now proceed with the proof.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">''Proof'': We first show that for any <math>n\in\mathbb{Z}^{+}</math>, every nonempty subset of <math>\{1,2,\ldots, n\}</math> has a smallest element.  Let <math>A</math> be the set of all positive integers <math>n</math> where this statement holds.  We see <math>A</math> contains <math>1</math>, since if <math>n=1</math> then the only subset of <math>\{1,2,\ldots, n\}</math> is <math>\{1\}</math> itself.  Then, supposing <math>A</math> contains <math>n</math> we show that it must contain <math>n+1</math>.  Let <math>C</math> be a nonempty subset of <math>\{1,2,\ldots, n+1\}</math>.  If <math>C=\{n+1\}</math> then <math>n+1</math> is its smallest element.  Otherwise, consider <math>C\cap \{1,2,\ldots,n\}</math>, which is nonempty.  Because <math>n\in A</math>, this set has a smallest element, which will be the smallest element of <math>C</math> also.  This means that <math>A</math> is inductive and <math>A=\mathbb{Z}^{+}</math>, so the statement is true for all <math>n\in\mathbb{Z}^{+}</math>.  </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins class="diffchange diffchange-inline">Now suppose <math>D\in \mathbb{Z}^{+}</math> is nonempty.  By choosing some <math>n\in D</math>, the set <math>A\cap\{1,2\ldots, n\}</math> is also nonempty which means that <math>A</math> has a smallest element <math>k</math>.  This means that <math>k</math> is the smallest element of <math>D</math> too, which completes the proof.</ins></div></td></tr>
</table>Arr0whttps://artofproblemsolving.com/wiki/index.php?title=Well_Ordering_Principle&diff=128339&oldid=prevAops turtle at 02:54, 15 July 20202020-07-15T02:54:37Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:54, 15 July 2020</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; 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>The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member.</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member<ins class="diffchange diffchange-inline">. The proof of this is simply common sense, but we can construct a formal proof by contradiction. Assume there is no smallest element. Then for each element in the set, there exists a smaller element, so if we take this smaller element, there must a different smaller element, and so on. Since the set is finite, we cannot continue like this infinitely many times, contradiction</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>{{stub}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>{{stub}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>[[Category:Axioms]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>[[Category:Axioms]]</div></td></tr>
</table>Aops turtlehttps://artofproblemsolving.com/wiki/index.php?title=Well_Ordering_Principle&diff=30368&oldid=prev1=2 at 15:57, 18 February 20092009-02-18T15:57:22Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member.</div></td></tr>
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</table>1=2https://artofproblemsolving.com/wiki/index.php?title=Well_Ordering_Principle&diff=30367&oldid=prev1=2: New page: The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member. {{stub}} Category:Axioms2009-02-18T15:57:15Z<p>New page: The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member. {{stub}} <a href="/wiki/index.php/Category:Axioms" title="Category:Axioms">Category:Axioms</a></p>
<p><b>New page</b></p><div>The '''Well Ordering Principle''' states that every nonempty set of positive integers contains a smallest member.<br />
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