User blog:Vel!/Ultracardinals

MD5:3EE0246B40614E1A47AFA041F4906FFD what?

Ultracardinals are a new class of cardinal numbers I invented. I believe they are far larger than Reinhardt cardinals, and may likely lead to the creation of a computable function even more powerful than any of those on this wiki.

A cardinal \(\kappa\) is an ultracardinal iff there exists no cardinal \(\alpha < \kappa\) such that the cardinality of the cofinality of \(\kappa\) may be expressed using 0, 1, \(\alpha\), addition, multiplication, and exponentiation, and the function \(C\) defined as \(C(\alpha) = \text{the cofinality of }\alpha\).

Ultracardinals are larger than Reinhardt cardinals, so they are inconsistent with the axiom of choice. However, not only are they inconsistent with ZFC, but their existence is also inconsistent with ZF's axiom of extensionality! However, unlike the Reinhardt cardinals, it is easy to confirm the existence of ultracardinals. We're unsure about whether Reinhardt cardinals exist, but we can be certain with ultracardinals.

Ultracardinals are strong enough that the Rathjen psi function fails to return any value with an ultracardinal as its input. We need to define a new extension to \(\psi\) in order for them to work. I call it the \(\mu\) function. If we let \(U\) be the smallest ultracardinal, then it is obvious that \(\nu(U)\) is much, much larger than the proof-theoretic ordinal of second-order arithmetic.

(to be continued)