User blog comment:Hyp cos/Question about weak compactness/@comment-35470197-20180911215601/@comment-35470197-20180912235451

> the ordinal has the "weakly compact property" over weakly compact cardinals less than it

You mean that an uncountable cardinal \(\kappa\) is "2-weakly compact" if it is weakly compact for any binary relation on it (to be precise, it here means a relation on \([\kappa]^2\)), there is a series of weakly compact cardinals converging to \(\kappa\) to which the restriction of either the relation or the negation is tautology, right?

Then the "2-weak compactness" is equivalent to the property that it is a weakly compact cardinal which is a limit of weakly compact cardinals.

Therefore every \(\Pi_2^1\)-indescribable cardinal is "2-weakly compact", because its subset of \(\Pi_1^1)-indescribable cardinals is stationary.

> Does the least \(\Pi_2^1\)-indescribable cardinal have all these weak compactness and large enough for collapsing over the weak compactness?

Maybe. I do not have a proof, though.

By the way, are you the author of the document you linked? The description of the notion of an ordinal notation looks strange. In the document, \(OT\) just forms a set of ordinals, but not a set of ordinal terms. It is recursively defined, but is not a recursive arithmetic system anymore. It strictly uses the \(\in\)-relation between ordinals. In order to obtain an ordinal notation system, one needs to define a recursive relation instead of \(\in\) as Buchholz and Rathjen explicitly did. Sorry if I misunderstood what is written.