Square principle

<h2 id="definition">Definition</h2>
<p>Define Sing to be the <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of all <a href="/facts/Limit_ordinal/jPCCoPuP">limit ordinals</a> which are not <a href="/facts/Regular_ordinal/wrGvPz2c">regular</a>.  <i>Global square</i> states that there is a system 
  
    
      
        (
        
          C
          
            β
          
        
        
          )
          
            β
            ∈
            
              S
              i
              n
              g
            
          
        
      
    
    {\displaystyle (C_{\beta })_{\beta \in \mathrm {Sing} }}
  
 satisfying:
</p>
<ol><li>
  
    
      
        
          C
          
            β
          
        
      
    
    {\displaystyle C_{\beta }}
  
 is a <a href="/facts/Club_set/TJVtvQAT">club set</a> of 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
.</li>
<li><a href="/facts/Order_type/koXjnax6">ot</a>
  
    
      
        (
        
          C
          
            β
          
        
        )
        <
        β
      
    
    {\displaystyle (C_{\beta })<\beta }
  
</li>
<li>If 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is a limit point of 
  
    
      
        
          C
          
            β
          
        
      
    
    {\displaystyle C_{\beta }}
  
 then 
  
    
      
        γ
        ∈
        
          S
          i
          n
          g
        
      
    
    {\displaystyle \gamma \in \mathrm {Sing} }
  
 and 
  
    
      
        
          C
          
            γ
          
        
        =
        
          C
          
            β
          
        
        ∩
        γ
      
    
    {\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }
  
</li></ol>
<h2 id="variant-relative-to-a-cardinal">Variant relative to a cardinal</h2>
<p>Jensen introduced also a local version of the principle.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> If

κ
      
    
    {\displaystyle \kappa }
  
 is an uncountable cardinal, 
then 
  
    
      
        
          ◻
          
            κ
          
        
      
    
    {\displaystyle \Box _{\kappa }}
  
 asserts that there is a sequence 
  
    
      
        (
        
          C
          
            β
          
        
        ∣
        β
        
           a limit point of 
        
        
          κ
          
            +
          
        
        )
      
    
    {\displaystyle (C_{\beta }\mid \beta {\text{ a limit point of }}\kappa ^{+})}
  
 satisfying:
</p>
<ol><li>
  
    
      
        
          C
          
            β
          
        
      
    
    {\displaystyle C_{\beta }}
  
 is a <a href="/facts/Club_set/TJVtvQAT">club set</a> of 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
.</li>
<li>If 
  
    
      
        c
        f
        β
        <
        κ
      
    
    {\displaystyle cf\beta <\kappa }
  
, then  
  
    
      
        
          |
        
        
          C
          
            β
          
        
        
          |
        
        <
        κ
      
    
    {\displaystyle |C_{\beta }|<\kappa }
  
</li>
<li>If 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is a limit point of 
  
    
      
        
          C
          
            β
          
        
      
    
    {\displaystyle C_{\beta }}
  
 then 
  
    
      
        
          C
          
            γ
          
        
        =
        
          C
          
            β
          
        
        ∩
        γ
      
    
    {\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }
  
</li></ol>
<p>Jensen proved that this principle holds in the constructible universe for any uncountable cardinal 
  
    
      
        κ
      
    
    {\displaystyle \kappa }
  
.
</p>
<h2 id="notes">Notes</h2>

<ul><li>Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy", <i>Annals of Mathematical Logic</i>, 4 (3): 229–308, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0003-4843%2872%2990001-0">10.1016/0003-4843(72)90001-0</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0309729">0309729</a></li></ul>

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

<ol>
<li id="fn:1"><p>Cummings, James (2005), "Notes on Singular Cardinal Combinatorics", Notre Dame Journal of Formal Logic, 46 (3): 251–282, doi:10.1305/ndjfl/1125409326 Section 4. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Jech, Thomas (2003), Set Theory: Third Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7, p. 443. <a href="978-3-540-44085-7" target="_blank">978-3-540-44085-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Square principle open-in-new

Square principle