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Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)

Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:

*Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B

*Type A = Type B if smallest Type A and Type B have same cardinalities

*Type A $\perp$ Type B if the ordering in the sense above is undecidable

So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo

What is known about the ordering of large cardinals from (http://cantorsattic.info/Upper_attic) in this sense ?

I am particularly interested in the "large" large cardinals from measurable upwards. For eg: How would one order measurable, extendible, huge and rank-into-rank ?

Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.

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    $\begingroup$ Just a note: of the three relations you define, one is not like the others. Your “>” and “=” are ordinary statements of set theory, while your “~” is a meta-statement about provability (in ZFC, I assume). The cognoscenti are used to this, but it’s helpful to be explicit about it: conversations can get very muddled when some participants miss this sort of issue. $\endgroup$ – Peter LeFanu Lumsdaine Sep 24 '15 at 7:54
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    $\begingroup$ The use of $\sim$ for undecidability seems particularly strange. How about $\bot$ instead? $\endgroup$ – Asaf Karagila Sep 24 '15 at 8:22
  • $\begingroup$ Related question: mathoverflow.net/questions/218414/… $\endgroup$ – Timothy Chow Sep 24 '15 at 20:33
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The usual relations to consider in the large cardinal hierarchy are

  • Direct implication: every A cardinal is also a B cardinal
  • Consistency strength implication: if ZFC + there is an A cardinal is consistent, then so is ZFC + there is a B cardinal.

Your concept, however, is focused on the least instance of the large cardinal notion, and this is also studied.

In broad terms, the large cardinal hierarchy is roughly linear, with the stronger cardinals being stronger with respect to all three of these relations. In most instances, we have that every A cardinal (the stronger notion) is also a B cardinal, as well as a limit of B cardinals, and so we get also the consistency implication and the least A cardinal is strictly larger than the least B cardinal.

However, there are some notable deviations from this. These deviations come in two types.

First, there are the instances where a large cardinal concept A has stronger consistency strength than B, but the least instance of A is definitely less than the least instance of B. For example, a superstrong cardinal has higher consistency strength than a mere strong cardinal, since if $\kappa$ is superstrong, then $V_\kappa\models$ ZFC + there is a proper class of strong cardinals, but the least superstrong cardinal is definitely less than the least strong cardinal. This is simply because superstrongness is witnessed by a single object, and strong cardinals are $\Sigma_2$ reflecting, and therefore reflect the least instance below.

There are numerous similar instances of this. Any time a large cardinal notion is witnessed by a single object or is witnessed inside some $V_\theta$ — and this would include weakly compact, Ramsey, measurable, superstrong, almost huge, huge, rank-to-rank and others — then the least instance of that cardinal will be less than the least $\Sigma_2$-reflecting cardinal and indeed less than the least $\Sigma_2$-correct cardinal. But $\Sigma_2$ correct cardinals provably exist in ZFC, and therefore have very low consistency strength.

So we have numerous interesting instances where your $<$ order does not align with consistency strength:

  • The least almost huge cardinal is strictly less than the least strong cardinal.
  • The least rank-to-rank cardinal is strictly less than the least strongly unfoldable cardinal.
  • The least $5$-huge cardinal is strictly less than the least uplifting cardinal.
  • There are hundreds of other similar examples. You can invent them yourself!

Meanwhile, second, there are examples of your $\perp$ situation, where the size of the smallest instance is not yet settled. This phenomenon is known as the "identity-crises" phenomenon, named by Magidor when he proved that the least measurable can be the same as the least strongly compact, or strictly less, depending on the model of set theory. Many further instances of this are now known, some of which appear in my paper:

This paper provides many instances of your $\perp$ situation, where the question of whether the least A cardinal is smaller than or the same size as the least B cardinal is not settled in ZFC.

Finally, let me qualify my remark that the large cardinal hierarchy is roughly linear. The hierarchy is indeed mainly linear, but one sometimes hears stronger assertions of linearity, as something that we know and which needs explanation, but I don't feel these knowledge claims are justified. Of course, the identity crises phenomenon provides instances of non-linearity in the direct implication hierarchy, and so when large cardinal set theorists assert that the large cardinal hierarchy is linear, they are speaking of the consistency strength order. So let me mention a few cases where we simply don't yet know linearity:

  • A supercompact cardinal versus a strongly compact plus an inaccessible above.

  • A supercompact cardinal versus a proper class of strongly compact cardinals.

  • A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal.

  • A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa^{++}$-strongly compact.

  • A PFA cardinal versus a strongly compact cardinal.

  • And many others.

My perspective is this. Because we have essentially no method for proving non-linearity in the consistency strength hierarchy, it is not surprising that we see only instances of linearity, and this may be a case of confirmation bias. But don't get me wrong: of course I agree that the consistency strength hierarchy is mainly linear in broad strokes.

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  • $\begingroup$ Fantastic answer and a lot to digest. Thank you very much. $\endgroup$ – Cosmonut Sep 24 '15 at 20:59
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Let me add one extra example that might be interesting.

Let $\pi_n^m$ and $\sigma_n^m$ denote respectively the least $\Pi_n^m$-indescribable and the least $\Sigma_n^m$-indescribable cardinal (if they exist). Then:

Fact 1. If $V=L,$ then $\sigma_n^m < \pi_n^m,$ for all $n, m \geq 1,$

Fact 2 (Hauser). If the existence of a $Σ^m_n$ indescribable above a $Π^m_n$ indescribable is consistent with $ZFC$, then the theory $ZFC+GCH+σ^m_n>π^m_n$ is consistent.

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