Axiom of projective determinacy

<h2 id="consequences">Consequences</h2>
<p>PD implies that all projective sets are <a href="/facts/Lebesgue_measurable/8us9cGoW">Lebesgue measurable</a> (in fact, <a href="/facts/Universally_measurable/8Utsfxle">universally measurable</a>) and have the <a href="/facts/Perfect_set_property/S8A7wxcN">perfect set property</a> and the <a href="/facts/Property_of_Baire/G2Sk5Azz">property of Baire</a>.  It also implies that every projective <a href="/facts/Binary_relation/r9BwGgaK">binary relation</a> may be <a href="/facts/Uniformization_(set_theory)/MLWcRIaz">uniformized</a> by a projective set.
</p><p>PD implies that for all positive integers 
  
    
      
        n
      
    
    {\displaystyle n}
  
, there is a largest countable 
  
    
      
        
          Σ
          
            2
            n
          
          
            1
          
        
      
    
    {\displaystyle \Sigma _{2n}^{1}}
  
 set.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>

<ul><li><a href="/facts/Donald_A._Martin/KsNsJnDP">Martin, Donald A.</a>; <a href="/facts/John_R._Steel/Twhy0kXJ">Steel, John R.</a> (Jan 1989). <a href="https://doi.org/10.2307%2F1990913">"A Proof of Projective Determinacy"</a>. <i><a href="/facts/Journal_of_the_American_Mathematical_Society/qR98drCm">Journal of the American Mathematical Society</a></i>. 2 (1): 71–125. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1990913">10.2307/1990913</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1990913">1990913</a>.</li>
<li><a href="/facts/Yiannis_N._Moschovakis/ajc4e46V">Moschovakis, Yiannis N.</a> (2009). <a href="https://web.archive.org/web/20141112111558/http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf"><i>Descriptive set theory</i></a> (PDF) (2nd ed.). Providence, R.I.: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4813-5. Archived from the original on 2014-11-12.{{cite book}}:  CS1 maint: bot: original URL status unknown (link)</li></ul>

<h3>Citations</h3>

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

<ol>
<li id="fn:1"><p>Donald A. Martin, "The largest countable this, that, and the other". Cabal seminar 79–81, Proceedings, Caltech-UCLA Logic Seminar 1979–81, edited by A. S. Kechris, D. A. Martin, and Y. N. Moschovakis, Lecture notes in mathematics, vol. 1019, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983, pp. 97–106. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Axiom of projective determinacy open-in-new

Axiom of projective determinacy