User blog comment:Hyp cos/Question about weak compactness/@comment-35470197-20180911215601/@comment-35470197-20180913113906

@Hyp cos

> That's quite weak.

Exactly. The reason why the "2-weak compactness" is so weak is because the "weakly compact property" for weakly compact cardinals below it just refers to the cardinality.

> For the use of OCF, one may need such property that

Such properties are desired, but are not necessarily needed. The properties just ensure the strength of the resulting OCF. For example, the least weakly Mahlo cardinal in the standard OCF with M can be replaced by \(\Lambda_0\).

In order to prove the well-definedness of an OCF \(\psi\) with normal form using a large cardinal axiom \(K\) as an extension of given finitely many enumeration functions \(\varphi,\chi,M, \ldots\) with normal forms, it suffices to show that cardinals satisfying \(A\) are not presented by iteration of \(\varphi,\chi,M, \ldots, \psi\) applied to ordinals below \(A\) so that every ordinal below a fixed limit admits a unique presentation by \(\varphi,\chi,M, \ldots, \psi\) of normal form.

But, yeah, I know that we want strength. Never mind it.

> Are there any other definition of "weakly compact property" suit this?

Sorry, I have not idea.

Beyond Mahlo's, the model theoretic properties (using elementary embeddings or indescribability) of the large cardinal axioms seem to be ones of the most useful aspects. Therefore it might be difficult to give "weak compactness" by observing series of ordinals.

> Certainly no.

Oops. I am sorry ;D

@Syst3ms

Your formulation is equivalent to mine. If \(K\) is 2-weakly compact in your sense, then \(\kappa\) is unbounded, i.e. \(K\) is a limit of weakly compact cardinals.

On the other hand, if \(K\) is a limit of weakly compact cardinals, then \(\kappa\) is of cardinality \(K\) by the regularity of \(K\), and hence \([\kappa]^2\) is bijective to \([K]^2\). Then the existence of a homogeneous subset follows from the weak compactness of \(K\) itself.

> With this definition, it is possible that a 2-weakly compact be suited for OCF purposes

Right. But the OP withes further strength.