About: Rank-into-rank cardinal   Sponge Permalink

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The rank-into-rank cardinals are uncountable cardinal numbers \(ho\) that satisfy one of these axioms: * I3. There exists a nontrivial elementary embedding \(j : V_ho \mapsto V_ho\). * I2. There exists a nontrivial elementary embedding \(j : V \mapsto M\), where \(V_ho \in M\) and \(ho\) is the first fixed point above the critical point of \(j\). * I1. There exists a nontrivial elementary embedding \(j : V_{ho + 1} \mapsto V_{ho + 1}\). * I0. There exists a nontrivial elementary embedding \(j : L(V_{ho + 1}) \mapsto L(V_{ho + 1})\)

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  • Rank-into-rank cardinal
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  • The rank-into-rank cardinals are uncountable cardinal numbers \(ho\) that satisfy one of these axioms: * I3. There exists a nontrivial elementary embedding \(j : V_ho \mapsto V_ho\). * I2. There exists a nontrivial elementary embedding \(j : V \mapsto M\), where \(V_ho \in M\) and \(ho\) is the first fixed point above the critical point of \(j\). * I1. There exists a nontrivial elementary embedding \(j : V_{ho + 1} \mapsto V_{ho + 1}\). * I0. There exists a nontrivial elementary embedding \(j : L(V_{ho + 1}) \mapsto L(V_{ho + 1})\)
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abstract
  • The rank-into-rank cardinals are uncountable cardinal numbers \(ho\) that satisfy one of these axioms: * I3. There exists a nontrivial elementary embedding \(j : V_ho \mapsto V_ho\). * I2. There exists a nontrivial elementary embedding \(j : V \mapsto M\), where \(V_ho \in M\) and \(ho\) is the first fixed point above the critical point of \(j\). * I1. There exists a nontrivial elementary embedding \(j : V_{ho + 1} \mapsto V_{ho + 1}\). * I0. There exists a nontrivial elementary embedding \(j : L(V_{ho + 1}) \mapsto L(V_{ho + 1})\) The four axioms are numbered in increasing strength: I0 implies I1, I1 implies I2, and I2 implies I3. A cardinal satisfying axiom I0 is called an I0 cardinal, and so forth. The axioms asserting the existence of the rank-into-ranks are extremely strong, so strong that there is good reason to doubt their consistency with ZFC. They are certainly not provable in ZFC (if it's consistent). If ZFC + "there exists a rank-into-rank cardinal" is consistent, then I0 rank-into-ranks are the largest kind of cardinals known that are compatible with ZFC. The existence of rank-into-ranks implies the divergence of the slow-growing \(p\) function from Laver tables (and the totality of its fast-growing pseudo-inverse \(q\)).
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