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The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By Reitz, it is first-order expressible and easy to force over any given ZFC model with class-sized forcing, preserving large cardinals.

Woodin showed that his Ultimate-L axiom implies the Ground Axiom. It doesn't seem to be "built in" to the statement of Woodin's axiom, so it is a very interesting result.

In contrast, forcing axioms do not imply the ground axiom because of preservation theorems. For example, PFA is preserved by $\omega_2$-closed forcing.

Question: Are there other set theoretic principles (that are natural, simple, or interesting) that imply the Ground Axiom, without being obviously designed to do so? Examples of non-fine-structural statements would be most interesting.

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    $\begingroup$ I suppose any principle implying the Ground Axiom is destroyed by every set-sized forcing. Already I think it's an interesting question to ask what set-theoretic principles, other than the obvious ones, are always destroyed by set-sized forcing. (The "obvious" ones, to me, are things like $V=L$ or $V=L[\mu]$.) $\endgroup$
    – Will Brian
    Commented Oct 25, 2023 at 14:27
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    $\begingroup$ This is perhaps a bit silly of an example, but axioms like $V = L$ or $V = L[0^\sharp]$ imply the ground axiom—the former because there are no (proper) inner models and the latter because $0^\sharp$ cannot be added by set forcing. $\endgroup$
    – Kameryn Williams
    Commented Oct 25, 2023 at 14:32
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    $\begingroup$ The ground axiom is due to Reitz and myself, as Jonas states in his paper. $\endgroup$
    – Joel David Hamkins
    Commented Oct 27, 2023 at 3:06
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    $\begingroup$ The first main theorems about it were proved by Jonas in his dissertation, including the fact that the CCA implies GA. Interestingly, he also proved that MA and PFA etc are consistent with the ground axiom. In particular, the common slogan that what those axioms assert is that a lot of forcing has already been done is not quite correct, if what is meant is set forcing. $\endgroup$
    – Joel David Hamkins
    Commented Oct 27, 2023 at 3:07
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    $\begingroup$ @JoelDavidHamkins I was responding to Will Brian's question about principles that are always destroyed by set forcing. I guess my comment was a bit cryptic, but the idea was to come up with a principle that is always destroyed by set forcing but does not imply the Ground Axiom, so I did mean minimal in the sense you suggested. (Note that the mantle is the only definable element of the set-generic multiverse. But my example shows there are nontrivial definable subsets of the multiverse. Maybe the definability theory here is interesting...) $\endgroup$
    – Gabe Goldberg
    Commented Oct 27, 2023 at 17:39

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The existence of a proper class of supercompact cardinals $\kappa$ that are indestructible by $\kappa$-directed closed forcing implies the Ground Axiom. This is because it implies the Continuum Coding Axiom, which implies the Ground Axiom.

You can actually get away with a proper class of indestructibly $\Sigma_2$-correct cardinals, a hypothesis with no consistency strength beyond ZFC. This principle has the following restatement that a certain kind of person might find philosophically compelling: if $a$ is a set and $\varphi$ is a $\Sigma_2$-formula such that $\varphi(a)$ can be forced by arbitrarily directed closed forcing, then $\varphi(a)$ is true.

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  • $\begingroup$ Thanks for your answer. Could you say a bit about the consistency proof for the class of indestructibly sigma2 correct cardinals? $\endgroup$
    – Monroe Eskew
    Commented Oct 27, 2023 at 5:01
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    $\begingroup$ The proof assuming a proper class of inaccessible $\Sigma_2$-reflecting cardinals (which follows from Ord is Mahlo) is in my paper "On Usuba's extendible cardinal," which is on the arXiv. In a more recent version of the paper, the assumption is reduced to just ZFC, but I'm still working on the presentation. I actually made every $\Sigma_2$-reflecting cardinal indestructible! $\endgroup$
    – Gabe Goldberg
    Commented Oct 27, 2023 at 14:59

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