Difference between revisions of "Middle attic"

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Welcome to the middle attic, where you will find many of the commonly used concepts of infinity in set theory. All cardinals here either provably exist or have existence equiconsistent with ZFC.
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{{DISPLAYTITLE: The middle attic}}
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[[File:StAugustineLighthouse.jpg | thumb | St. Augustine Lighthouse photo by Madrigar]]
  
[[File:StAugustineLighthouse.jpg | right | St. Augustine Lighthouse photo by Madrigar]]
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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.
  
* Up to the [[upper attic]]
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* into the [[upper attic]]
* the [[reflecting#Feferman theory | Feferman theory]]
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* [[correct]] cardinals, [[reflecting | $V_\delta\prec V$]] and the [[reflecting#Feferman theory | Feferman theory]]
* [[reflecting | $\Sigma_n$-reflecting]] and the fully [[reflecting]] cardinals $V_\delta\prec V$
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* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$ correct]] and [[reflecting | $\Sigma_n$-correct]] cardinals
* [[reflecting#Sigma2 reflecting | $\Sigma_2$ reflecting cardinal]]
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* [[extendible#-extendible cardinals | 0-extendible]] cardinal
* [[beth hierarchy#beth_fixed_point | $\beth$-fixed point]]
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* [[extendible#Sigma_n-extendible cardinals|$\Sigma_n$-extendible]] cardinal
* [[strong limit]] cardinal
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* [[beth#beth_fixed_point | $\beth$-fixed point]]
* the [[beth hierarchy | $\beth_\alpha$ hierarchy]]
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* the [[beth]] numbers and the [[beth | $\beth_\alpha$ hierarchy]]
* [[aleph hierarchy#aleph_fixed_point | $\aleph$-fixed point]]
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* [[beth omega | $\beth_\omega$]] and the [[strong limit]] cardinals
* the [[aleph hierarchy | $\aleph_\alpha$ hierarchy]]
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* [[Theta | $\Theta$]]
* [[aleph omega | $\aleph_\omega$]] and [[regular#singular | singular]] cardinals
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* the [[continuum]]
 
* the [[continuum]]
* [[regular]] cardinals
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* [[cardinal characteristics]]  of the continuum
* [[Aleph_1 | $\aleph_1$]], the first uncountable cardinal
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** 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
* [[Omega | $\aleph_0$]] and the rest of the [[lower attic]]
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* the [[descriptive set theory | descriptive set-theoretic]] cardinals
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* [[aleph#aleph fixed point | $\aleph$-fixed point]]
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* the [[aleph]] numbers and the [[aleph | $\aleph_\alpha$ hierarchy]]
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* [[Buchholz's ψ functions]]
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* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals
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* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal
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* [[uncountable]], [[regular]] and [[successor]] cardinals
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* [[aleph#aleph one | $\aleph_1$]], the first [[uncountable]] cardinal
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* [[cardinal | cardinals]], [[infinite]] cardinals
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* [[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.