By Gro-Tsen

2019-01-08 12:18:35 8 Comments

Consider the following two observations:

  1. The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals.

  2. The axiom of Choice is incompatible with large cardinal axioms that are somehow "too large" (much larger than in the previous point), like Reinhardt cardinals.

I would like to understand whether these two observations should be thought of as being of a similar nature. (I note that the first seems to be construed as an argument against $V=L$ because it limits the size of possible cardinals, whereas the second does not seem to have been used much as an argument against accepting Choice as an axiom. This could be for historical/sociological reasons, perhaps because Choice is too useful in ordinary mathematical practice, or because the parallel I try to draw between (1) and (2) is flawed.) As such, the question is probably too vague to ask here. Instead, let me ask:

Are there any known and interesting combinatorial principles of a related nature that are incompatible with "too large" cardinals which could sit alongside the above observations?

This is still rather vague, of course, because I don't know what kind of combinatorial principles can be considered related to $V=L$ and Choice, but it could be:

  • Beyond (1): principles that are even stronger than $V=L$ (i.e., imply it) and which exclude cardinals at an even smaller size than measurable ones. (This would be very interesting, but I don't think there is any reasonable combinatorial principle known to imply $V=L$.)

  • Between (1) and (2): consequences of $V=L$ (implying Choice, or assuming Choice alongside) which exclude certain very large cardinals but are still compatible with measurable cardinals. I expect convincing examples of this can be given.

  • Beyond (2) but still in $\mathsf{ZF}$: weak forms of Choice that are believed to be compatible with Reinhardt cardinals but still incompatible with some even larger cardinals thought to be consistent with $\mathsf{ZF}$.

  • Even beyond $\mathsf{ZF}$: maybe it makes sense to consider some part of $\mathsf{ZF}$ itself as a "combinatorial principle" and formulate even larger cardinals that are compatible with weaker set theories although not with $\mathsf{ZF}$? (Perhaps the law of the excluded middle can be considered in the line of $V=L$ and Choice?)

Of course, the "combinatorial principle" has to have some kind of useful content to it in structuring or ordering the set-theoretic Universe, not "there does not exist a supercompact cardinal".


@Joel David Hamkins 2019-01-08 14:29:48

Consider the relativized constructibility hypothesis, which asserts that $V=L[A]$ for some set $A$.

This axiom is compatible with any locally verifiable large cardinal property, properties that can be witnessed by a certain fact inside some sufficiently large $V_\theta$. See my blog post, local properties in set theory for more discussion of this, including the fact that the locally verifiable properties are precisely the $\Sigma_2$ properties.

Many large cardinal properties are locally verifiable, including: weak compactness, Ramsey, subtle, measurable, measurable with specified $o(\kappa)$, superstrong, almost huge, huge, I0, I1, I2, and others. These notions span a huge part of the large cardinal hierarchy. All these notions are relatively consistent with the hypothesis $\exists A\ V=L[A]$.

Meanwhile, many other large cardinal properties are not locally verifiable, since they require one to have witnesses for arbitrarily large ordinals, making the properties $\Pi_3$ rather than $\Sigma_2$. For example, reflecting cardinal, strong, strongly compact, supercompact, extendible, superhuge and others. None of these notions is consistent with $\exists A\ V=L[A]$.

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