Successor ordinal

<h2 id="properties">Properties</h2>
<p>Every ordinal other than 0 is either a successor ordinal or a <a href="/facts/Limit_ordinal/jPCCoPuP">limit ordinal</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="in-von-neumanns-model">In Von Neumann's model</h2>
<p>Using <a href="/facts/Ordinal_number/GelFLjJt">von Neumann's ordinal numbers</a> (the standard model of the ordinals used in set theory), the successor <i>S</i>(<i>α</i>) of an ordinal number <i>α</i> is given by the formula<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>

S
        (
        α
        )
        =
        α
        ∪
        {
        α
        }
        .
      
    
    {\displaystyle S(\alpha )=\alpha \cup \{\alpha \}.}

<p>Since the ordering on the ordinal numbers is given by <i>α</i> < <i>β</i> if and only if <i>α</i> ∈ <i>β</i>, it is immediate that there is no ordinal number between α and <i>S</i>(<i>α</i>), and it is also clear that <i>α</i> < <i>S</i>(<i>α</i>).
</p>
<h2 id="ordinal-addition">Ordinal addition</h2>
<p>The successor operation can be used to define <a href="/facts/Ordinal_arithmetic/satc9FHy">ordinal addition</a> rigorously via <a href="/facts/Transfinite_induction/XlolgTDl">transfinite recursion</a> as follows:
</p>

α
        +
        0
        =
        α
        
      
    
    {\displaystyle \alpha +0=\alpha \!}

α
        +
        S
        (
        β
        )
        =
        S
        (
        α
        +
        β
        )
      
    
    {\displaystyle \alpha +S(\beta )=S(\alpha +\beta )}

<p>and for a limit ordinal <i>λ</i>
</p>

α
        +
        λ
        =
        
          ⋃
          
            β
            <
            λ
          
        
        (
        α
        +
        β
        )
      
    
    {\displaystyle \alpha +\lambda =\bigcup _{\beta <\lambda }(\alpha +\beta )}

<p>In particular, <i>S</i>(<i>α</i>) = <i>α</i> + 1. Multiplication and exponentiation are defined similarly.
</p>
<h2 id="topology">Topology</h2>
<p>The successor points and zero are the <a href="/facts/Isolated_point/d3EeWxuD">isolated points</a> of the class of ordinal numbers, with respect to the <a href="/facts/Order_topology/EHISytz6">order topology</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ordinal_arithmetic/satc9FHy">Ordinal arithmetic</a></li>
<li><a href="/facts/Limit_ordinal/jPCCoPuP">Limit ordinal</a></li>
<li><a href="/facts/Successor_cardinal/NmowSfwc">Successor cardinal</a></li></ul>

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

<ol>
<li id="fn:1"><p>Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569. <a href="9781852330569" target="_blank">9781852330569</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569. <a href="9781852330569" target="_blank">9781852330569</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory, Undergraduate Texts in Mathematics, Springer, Exercise 3C, p. 100, ISBN 9780387940946. <a href="9780387940946" target="_blank">9780387940946</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Successor ordinal open-in-new

Successor ordinal