## FANDOM

10,828 Pages

1)

$$B_0(\alpha,\beta) = \beta \cup \{0,1,\omega,\Omega,\Omega^+\}$$

$$B_{n+1}(\alpha,\beta) = \{\gamma<\Omega^{++}:\text{cof}(\gamma) \in B_n(\alpha,\beta)\} \cup \{\Psi(\gamma):\gamma \in B_n(\alpha,\beta)\cap\alpha\}$$

$$B(\alpha,\beta) = \cup_{n<\omega}B_n(\alpha,\beta)$$

$$\Psi(\alpha) = \min\{\beta<\Omega:\beta\notin B(\alpha,\beta)\}$$

2)

Let $$V[G_{\kappa \leftarrow \lambda}]$$ be the Levy collapse of $$\lambda$$ onto $$\kappa$$, for regular $$\kappa$$, $$\lambda$$. Let $$B <^* A$$ denote that all limit points of $$B$$ are in $$A$$.

$$o(A) = \sup\{o(B)+1: B <^* A; B \subseteq \sup A\}$$

$$o(\alpha) = \sup\{o(A)+1 : A \subseteq \alpha \cap \text{Reg}; \sup A = \alpha\}$$

$$\Psi_\Omega(\alpha) = \min\{\beta<\Omega:(o(\beta))^{V[G_{\Omega \leftarrow |\alpha|^+}]} \geq \alpha\}$$

3)

This bit isn't a notation. If I'm correct, the two above notations (previously undiscovered) have the potential to greatly simplify ordinal analyses above those that typically collapse a weakly compact cardinal. A greatly Mahlo cardinal can be used in its place, and if I'm correct this clocks in at a mere $$\Psi((\Omega^+)^2)$$ in both notations. (The two notations aren't identical, but I think they catch up at multiples of $$\Omega^\omega$$, so few worries there.)

The key is that we can collapse all ordinals (not just the ones we can express in our notation) up to $$\Omega^+$$ (which are various levels of inaccessibility), hence allowing $$\Omega^+$$ to actually, successfully, collapse to the least weakly Mahlo cardinal, and successive additions of $$\Omega^+$$ corresponding to higher tiers of Mahloness. This goes in a different, simpler, direction compared to most OCFs that pass this point, and I hope it is accessible to the community here.

I haven't included the functions to collapse below regular cardinals as they are effectively clutter to the purpose of this post; if someone wants to make a post for them, possibly explaining this stuff better, they are welcome.

(PS: It is theoretically possible that the types of large cardinals required are inconsistent, although this seems unlikely given likely similarities to the Mitchell order. And no, I haven't proved stuff; I probably would have pushed this out much sooner if I hadn't bothered trying.)

PPS: Hnng. Can't change the title (to 'A new sort of OCF', or anything else).