Serial relation

<h2 id="russells-series">Russell's series</h2>
Relations are used to develop series in The Principles of Mathematics. The prototype is Peano's <a href="/facts/Successor_function/R1hhkSEG">successor function</a> as a one-one relation on the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>. Russell's series may be finite or generated by a relation giving <a href="/facts/Cyclic_order/j5DaU26j">cyclic order</a>. In that case, the <a href="/facts/Point-pair_separation/x18E5TWK">point-pair separation</a> relation is used for description. To define a progression, he requires the generating relation to be a <a href="/facts/Connected_relation/UpwopAts">connected relation</a>. Then <a href="/facts/Ordinal_number/GelFLjJt">ordinal numbers</a> are derived from progressions, the finite ones are finite ordinals.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: Chapter 28: Progressions and ordinal numbers  Distinguishing open and closed series<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>: 234  results in four total orders: finite, one end, no end and open, and no end and closed.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>: 202 
Contrary to other writers, Russell admits negative ordinals. For motivation, consider the scales of <a href="/facts/Measurement/rsLLbn1s">measurement</a> using <a href="/facts/Scientific_notation/YgyYEmq8">scientific notation</a>, where a power of ten represents a <a href="/facts/Decade_(log_scale)/qEYTPWw8">decade</a> of measure. Informally, this parameter corresponds to <a href="/facts/Orders_of_magnitude/cWzRP2dM">orders of magnitude</a> used to quantify physical units. The parameter takes on negative as well as positive values.

<h3>Stretch</h3>
Russell adopted the term stretch from <a href="/facts/Alexius_Meinong/AIrFc5vh">Meinong</a>, who had contributed to the theory of distance.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole."<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>: 181  To explain Meinong, Russell refers to the <a href="/facts/Cayley%E2%80%93Klein_metric/ky7Gn163">Cayley–Klein metric</a>, which uses stretch coordinates in <a href="/facts/Cross_ratio/tj4DK3Vk">anharmonic ratios</a> which determine distance by using logarithm.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>: 255 <a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h2 id="external-links">External links</h2>
<ul><li>Jing Tao Yao and Davide Ciucci and Yan Zhang (2015). <a href="https://books.google.com/books?id=gLS4CQAAQBAJ&pg=PA416">"Generalized Rough Sets"</a>. In Janusz Kacprzyk and Witold Pedrycz (ed.). Handbook of Computational Intelligence. Springer. pp. 413–424. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783662435052. Here: page 416.</li>
<li>Yao, Y.Y.; Wong, S.K.M. (1995). <a href="http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf">"Generalization of rough sets using relationships between attribute values"</a> (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858. <a href="978-1-136-76573-5" target="_blank">978-1-136-76573-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">B. A. Bernstein (1926) "On the Serial Relations in Boolean Algebras", Bulletin of the American Mathematical Society 32(5): 523,524 <a href="/wiki/B._A._Bernstein" target="_blank">/wiki/B._A._Bernstein</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Yao, Y. (2004). "Semantics of Fuzzy Sets in Rough Set Theory". Transactions on Rough Sets II. Lecture Notes in Computer Science. Vol. 3135. p. 309. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1. <a href="978-3-540-23990-1" target="_blank">978-3-540-23990-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">James Garson (2013) Modal Logic for Philosophers, chapter 11: Relationships between modal logics, figure 11.1 page 220, Cambridge University Press doi:10.1017/CBO97811393421117.014 <a href="/wiki/James_Garson" target="_blank">/wiki/James_Garson</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858. <a href="978-1-136-76573-5" target="_blank">978-1-136-76573-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858. <a href="978-1-136-76573-5" target="_blank">978-1-136-76573-5</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858. <a href="978-1-136-76573-5" target="_blank">978-1-136-76573-5</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Alexius Meinong (1896) Uber die Bedeutung der Weberische Gesetze <a href="/wiki/Alexius_Meinong" target="_blank">/wiki/Alexius_Meinong</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858. <a href="978-1-136-76573-5" target="_blank">978-1-136-76573-5</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Russell, Bertrand. Principles of mathematics. ISBN 978-1-136-76573-5. OCLC 1203009858. <a href="978-1-136-76573-5" target="_blank">978-1-136-76573-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Russell (1897) An Essay on the Foundations of Geometry <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Serial relation open-in-new

Serial relation