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.

Call a cardinal \(\kappa\) a "weak 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 \(C(\alpha) = \text{the cofinality of }\alpha\). Then, \(\sigma\) is an ultracardinal if there is no weak ultracardinal \(\kappa > \sigma\) such that

\[\kappa \geq \sup\{C(\alpha), \alpha^\beta: \alpha, \beta < \kappa\}.\]

(I'm still working on this definition, so it's subject to change.)

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, by the following (incomplete) definition:

\begin{eqnarray*} A_0(\alpha, \beta) &=& \beta \cup \{0, 1, \omega, \Omega\}\\ A_{n+1}(\alpha, \beta) &=& \{\gamma + \delta, \omega^{\gamma}, \mu(\eta), C(\gamma) | \gamma, \delta, \eta \in A_n (\alpha, \beta); \eta < \alpha\} \\ A(\alpha, \beta) &=& \bigcup_{n < \omega} A_n (\alpha, \beta) \\ \mu(\alpha) &=& \min \{\beta < \Omega | A(\alpha, \beta) \cap \Omega_{A(\alpha, \beta)} \subseteq \beta \wedge \alpha \in A(\alpha, \beta)\} \\ \end{eqnarray*}

If we let \(U\) be the smallest ultracardinal, then it is obvious that \(\mu(U)\) is much, much larger than the proof-theoretic ordinal of second-order arithmetic.

But we can go further than \(U\) by devising higher-order ultracardinals. If we dive into the definition and redefine \(C(\alpha) = \text{the order-2 cofinality of }\alpha\), we get the order-2 ultracardinals, the smallest of which is \(U_2\). The \(\mu\) function must be 2-argument in order for us to collapse these cardinals. Extensions are fairly intuitive:

  • \(\mu(U)\)
  • \(U_2 = \mu(U, 2)\)
  • \(U_3 = \mu(U, 3)\)
  • \(U_4 = \mu(U, 4)\)
  • \(U_{U_{U_{._{._.}}}} = \mu(U, \Omega)\)
  • \(\mu(U, U)\)
  • \(\mu(U, U, U)\)
  • \(\mu(U, U, U, \ldots) = \mu(U, \omega (1) 2)\)
  • \(\mu(U, \omega (0, 1) 2)\)
  • \(\mu(U, \omega (U) 2)\)
  • etc.

Hyperultracardinals an beyond

Most of this section is an improvement on Aarex's comment below, which (surprisingly for Aarex) is actually fairly legitimate. However, formalization of his ideas is still under way.

The limit of all this is the first hyperultracardinal \(H\). \(H\) is the first cardinal inexpressible as the limit of limit of limit of ... of limit of order-\(\alpha\) ultracardinals. We have the following hierarchy above \(H\):

  • \(H_2 = \mu_0(H, 2)\)
  • \(H_3 = \mu_0(H, 3)\)
  • \(H_{H_{H_{._{._.}}}} = \mu_0(H, \Omega)\)
  • \(\mu_0(H, \Omega_2)\)
  • \(\mu_0(H, \Omega_\omega)\)
  • \(\mu_0(H, \Omega_\Omega)\)
  • \(\mu_0(H, U)\)
  • \(\mu_0(H, H)\)
  • \(\mu_0(H, H (1) 2)\)
  • \(\mu_0(\{LL...LLH,H\}_{H,H}\) w/ H L's
  • etc.

The limit of all this is \(M\), the first megaultracardinal. Similarly, the limit of all the megaultracardinals and their extensions are the gigaultracardinals (the smallest of which is \(G\)).