# Ground Axiom and behaviors of continuum function

The Ground Axiom ($GA$) is the assertion that the universe of sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing $P\in W$.

Is $GA$ consistent with any possible behavior of continuum function $\kappa\mapsto 2^{\kappa}$?

It seems in models of $GA$ like $L$ and some other canonical models the growth speed of continuum function is too low (e.g. $L\models GCH$). So the natural question is:

What is the consistency situation for faster growth speeds of $\kappa\mapsto 2^{\kappa}$?

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In the paper The ground axiom is consistent with $V\ne{\rm HOD}$ (J. D. Hamkins, J. Reitz, W.H. Woodin, PAMS 136(8):2008), we prove that the ground axiom is consistent with $V\neq\text{HOD}$, and remark there that:

The proof of Theorem 1 is flexible and generalizes in a variety of ways. For example, not much was used about the specific iteration $\mathbb{P}$. For establishing GA, we needed to know only that the stage $\gamma$ forcing was ${\lt}\gamma$-closed, and we could easily have accommodated $\text{Add}(\gamma,\gamma^{++})$ or occasionally $\text{Coll}(\gamma,\gamma^{+})$, for example, without any difficulty in the argument. Also, we needn't have forced specifically at every regular cardinal stage $\gamma$, but could have forced at regular cardinals in some other unbounded pattern. Thus, the argument establishes that after forcing over $L$ with any of the usual reverse Easton iterations of closed forcing, one obtains the Ground Axiom in the extension.

Thus, with this kind of forcing, one can achieve a wide spectrum of patterns in the continuum function. Although the usual Easton forcing itself is a product, rather than an iteration, and will therefore definitely not result in the ground axiom, nevertheless one may transform the usual Easton product into an iteration, by taking big chunks of the forcing at a time. That is, given an Easton function $E$, one looks at sufficient closure points of $E$, and performs the iteration, which performs the usual Easton product between these closure points. The result will be a model of GA with the desired Easton function as the continuum function. I don't know if the details of this argument, however, have ever been written down, and if you were inclined in that direction, I would encourage you to do it (feel free to contact me).

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Sorry, I fixed the typo (thanks!). The linked paper proves the consistency with V$\neq$HOD. The consistency of GA with V=HOD was already known in the dissertation of Jonas Reitz, who proved that GA is a consequence of the Continuum Coding Axiom, which is a strong version of V=HOD. We didn't at that time know how to force GA except by this strong coding method, and the paper with Woodin solved that problem by providing an alternative way to prove GA. The end result is that almost any class length iteration of progressively closed forcing will force GA. –  Joel David Hamkins Mar 10 at 16:15
Ok, so you just had a typo (which I see you've fixed). Thanks for the clarification! –  Nate Eldredge Mar 10 at 16:16
An interesting tidbit: the authors of that paper constitute three mathematical generations, since the third author was the Ph.D. supervisor of the first, who was the Ph.D. supervisor of the second. –  Joel David Hamkins Mar 10 at 16:23
So is it true to say the Ground Axiom doesn't carry any information about the behavior of continuum function or there are some (few) known inconsistencies between some non-trivial values of continuum function and $GA$? –  user47697 Mar 10 at 17:03
I don't know of any inconsistencies between GA and the values of the continuum, and the argument I mentioned in my answer suggests that one can get GA with any Easton function on the regular cardinals (but as I said one should still check all the details). –  Joel David Hamkins Mar 10 at 17:12