# Re: Paper and slides on indefiniteness of CH

Dear Bill,

On Thu, 9 Oct 2014, William Tait wrote:

I think that somewhere in the vast literature in this thread piling up in my email box is the claim by Sy that the existence of cardinals up to Erdos $\kappa_{\omega_1}$ (or something like that) is implicit in the notion of the iterative hierarchy.

No, Erdos $\kappa_{\omega_1}$ gives you the existence of $0^\#$ and my position is that Height-Maximality is consistent V = L.

That does seem to me to be questionable; at least from the “bottom-up” point of view that what can be intrinsically justified are engines for iterating. Sy, as I recall, believes that axioms are intrinsically justified by reference to the universe V that they describe—a top-down point of view. No slander intended, Sy!

And no slander committed! I analyse Height Maximality as a height potentialist, comparing V to its lengthenings and shortenings. A close reading of my e-mails reveals an unannounced move on my part to distinguish Reflection from more powerful forms of Height Maximality (I arrive at #-generation, which implies the existence of all large cardinals compatible with V = L but does not imply the existence of $0^\#$!).

So I am no longer challenging the limits of Reflection, but instead asserting that Height Maximality is much more than that.

Best,
Sy

PS: Today I fly to California for a week of “real set theory” (I hope we won’t talk about truth there!). My aim is to drop out of this discussion whilst there, as the collaboration will be all-consuming and leave no time for anything else. But maybe I’ll type out another e-mail or two to this group in case I am awake at some weird hour due to jet-lag.