Suppose I want to prove some statement S is independent of ZFC.
Now instead of the usual approach of making models, I do the following:
- Take two large cardinal axioms L1 and L2
- Prove that ZFC + L1 $\implies$ S is true
- Prove that ZFC + L2 $\implies$ S is false
Then I argue that "Since ZFC + L1 implies S is true, then this means ZFC cannot prove S is false. Similarly, I have shown that ZFC cannot prove that S is true. Hence, S is independent of ZFC."

My questions are:
1) Does this approach work ? If so, has it actually been used ?
2) If not, what is the problem ? Is it the case that proving "S is consistent with ZFC" different from proving that "ZFC cannot prove S is false" ?
3) Or is the worry that I am adding axioms which might themselves be inconsistent with ZFC ? I realize that L1 and L2 could have been any two axioms, but I deliberately mentioned large cardinal axioms as they are generally believed to be consistent with ZFC.

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    $\begingroup$ If you could prove both implications then ZFC + L1 and ZFC + L2 would be inconsistent with each other. But any two of the standard large cardinal axioms are known to be consistent with each other, assuming each is individually consistent. Often one proves "S is true" from V=L and "S is false" from a large cardinal axiom. $\endgroup$ – Nik Weaver Jul 25 '19 at 4:29
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    $\begingroup$ I don't really know of any two incompatible large cardinal axioms. So a proof like that would be quite fantastic, and would either show that large cardinals are inconsistent, or come up with entirely new and different large cardinals. $\endgroup$ – Asaf Karagila Jul 25 '19 at 5:57
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    $\begingroup$ Though there are "fake" large cardinal axioms (known as \emph{ideal axioms}), asserting the existence of ideals satisfying certain properties, that are inconsistent with each other. $\endgroup$ – Otto Jul 25 '19 at 12:46
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    $\begingroup$ @Otto I wouldn't call these "fake large cardinal axioms" any more than PFA is a "fake large cardinal axiom". These are set theoretic axioms which has consistency strength that requires/implies the consistency of some large cardinal axioms in inner/outer models. $\endgroup$ – Asaf Karagila Jul 25 '19 at 13:12

The fundamental flaw with this approach is this part:

$\text{-}$ Take two large cardinal axioms L1 and L2

There is no singular formal definition of large cardinals, but in general, the fundamental property of large cardinals is:

  1. They form a linear hierarchy.

  2. Any two large cardinals are compatible with each other, assuming a sufficiently large large cardinal.

Here is the intuition behind 2. Given any large cardinal $\text{L}1$, $\text{L}1(\kappa)$ fundamentally asserts that $\kappa$ is so big that "[insert axiom]." It seems strange that there could be a number so big that "[insert axiom 1]," and there can be a number so big that "[insert axiom 2]," but not both at the same time.

On the other, hand there is a technique that goes like this: Take a large cardinal $\text{L}1$ and an axiom $S$. Then:

  1. $\text{L1}\rightarrow\text{Con}(\text{ZFC}+S)$

  2. $\text{L}1\rightarrow\neg S$

$S$ is the axiom $\exists U(V=L[U]\land U\text{ is a measure over a measurable cardinal})$. Let $\text{L}1$ be the assertion that there are two measurable cardinals. Then, by taking a witness to $0^\dagger$ exists, we can get a model $L[U]=V^{L[U]}$. But, because there are two measurable cardinals, $V\neq L[U]$.

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    $\begingroup$ They form a linear hierarchy? That's good to know! Which one comes first, then, superstrong or strongly compact? More to the point, the claim about linearity could be wrong, and it might be that it's linear because we're walking on a linear path which pushes us to keep adding things "in between existing axioms" that preserve linearity. That's not a "fundamental property", in that case, but rather an "accidental limitation of our method". $\endgroup$ – Asaf Karagila Nov 9 '19 at 7:35
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    $\begingroup$ Also, we have no tools to prove instances of nonlinearity in consistency strength. $\endgroup$ – Joel David Hamkins Nov 9 '19 at 13:17
  • $\begingroup$ @AsafKaragila Notice I never said that we know of the exact order in every case. It probably is the case that it is linear because we're walking on a linear path, but is that really a problem? That path is the large cardinal hierarchy. It is like saying "Not all towers are straight. It is just that we build all our towers to be straight." We have defined "towers" as one of the straight things we build, and a non-straight tower is a flaw in the tower, not in the definition of tower. More to the point, there are hundreds of large cardinals [Cont] $\endgroup$ – Master Nov 9 '19 at 16:08
  • $\begingroup$ Thousands if we count all the little variants that have been invented on mathoverflow posts and blogs. Despite this, while we sometimes do not know the exact order, we have never found a non-linearity, at least to my knowledge. Occam's razor dictates that this is probably because the hierarchy of large cardinals is linear. $\endgroup$ – Master Nov 9 '19 at 16:10
  • $\begingroup$ But that's the thing: saying that being straight is a fundamental property of a tower is not right. The fundamental property of a tower is that it is a tall and narrow building. Lacking the tools to construct non-linear large cardinal axioms is not a reason to postulate that they are "fundamentally linear". It's the same thing as saying that a language is "fundamentally a way for humans to communicate", which makes it impossible for any animals to have a language. It's just an unnecessary restriction. $\endgroup$ – Asaf Karagila Nov 9 '19 at 17:34

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