Difference between revisions of "Middle attic"

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Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.
 
Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.
  
* Up to the [[upper attic]]
+
* into the [[upper attic]]
* the [[reflecting#Feferman theory | Feferman theory]]
+
* [[correct]] cardinals, [[reflecting | $V_\delta\prec V$]] and the [[reflecting#Feferman theory | Feferman theory]]
* [[reflecting | $\Sigma_n$-reflecting]] and the fully [[reflecting]] cardinals $V_\delta\prec V$
+
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$ correct]] and [[reflecting | $\Sigma_n$-correct]] cardinals
* [[reflecting#Sigma2 reflecting | $\Sigma_2$ reflecting cardinal]]
+
* [[extendible#-extendible cardinals | 0-extendible]] cardinal
 +
* [[extendible#Sigma_n-extendible cardinals|$\Sigma_n$-extendible]] cardinal
 
* [[beth#beth_fixed_point | $\beth$-fixed point]]
 
* [[beth#beth_fixed_point | $\beth$-fixed point]]
* [[beth omega]] and the [[strong limit]] cardinals
+
* the [[beth]] numbers and the [[beth | $\beth_\alpha$ hierarchy]]
* the [[beth | $\beth_\alpha$ hierarchy]]
+
* [[beth omega | $\beth_\omega$]] and the [[strong limit]] cardinals
 +
* [[Theta | $\Theta$]]
 
* the [[continuum]]
 
* the [[continuum]]
 +
* [[cardinal characteristics]]  of the continuum
 +
** the [[bounding number | bounding number $\frak{b}$]], the [[dominating number | dominating number $\frak{d}$]], the [[covering number | covering numbers]], [[additivity number | additivity numbers]] and many more
 +
* the [[descriptive set theory | descriptive set-theoretic]] cardinals
 
* [[aleph#aleph fixed point | $\aleph$-fixed point]]
 
* [[aleph#aleph fixed point | $\aleph$-fixed point]]
* the [[aleph | $\aleph_\alpha$ hierarchy]]
+
* the [[aleph]] numbers and the [[aleph | $\aleph_\alpha$ hierarchy]]
 +
* [[Buchholz's ψ functions]]
 
* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals
 
* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals
 
* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal
 
* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal
* [[aleph#successor cardinals | successor cardinals]]
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* [[uncountable]], [[regular]] and [[successor]] cardinals
* [[regular]] cardinals
+
* [[aleph#aleph one | $\aleph_1$]], the first [[uncountable]] cardinal
* [[aleph#aleph one | $\aleph_1$]], the first uncountable cardinal
+
* [[cardinal | cardinals]], [[infinite]] cardinals
* [[Omega | $\aleph_0$]] and the rest of the [[lower attic]]
+
* [[Aleph zero | $\aleph_0$]] and the rest of the [[lower attic]]

Latest revision as of 07:24, 19 May 2018

St. Augustine Lighthouse photo by Madrigar

Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.