User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-5150073-20140515153351/@comment-5529393-20140605095447

Ikosarakt1: K does not represent $$\Xi(1)$$, it is a much larger cardinal. If by diagonalizer of $$\Xi(\alpha)$$ you mean the diagonalizer of $$\Psi_{Xi(\alpha}$$, that is $$\Xi(\alpha)$$ itself; if by diagonalizer of $$\Xi(\alpha)$$ you mean a cardinal that collapses to fixed points of $$\Xi(\alpha)$$ and its extensions, we can call that cardinal $$\Xi(1,0)$$. We can extend that to $$\Xi(\alpha,\beta)$$ for arbitrary $$\alpha, \beta$$. Then $$\Xi(\alpha, 0)$$ is diagonalized by $$\Xi(1,0,0)$$. And so on for more and more variables.

So what is K? K is used in a Bachmann-Howard-like extension of the $$\Xi$$, just as $$\Omega$$ is used in the Bachmann-Howard notation to extend the Extended Veblen notation.

So for example $$\Xi(\alpha, \beta, \gamma) = \Xi(K^2 \alpha + K \beta + \gamma)$$.