Ordering of large cardinals by cardinality

Question

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.

Answer 1

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:

• B. Cody, M. Gitik, J. D. Hamkins, and J. A. Schanker, The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact, Archive for Mathematical Logic, pp. 1-20, 2015.

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.

Answer 2

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.

Answer 3

To elaborate on Joel David Hamkins's answer: When the the size order of large cardinal properties differs from the strength order and it is not an example of the identity crisis phenomenon, it is usually because the weaker large cardinal notion implies stronger correctness properties. If a large cardinal property has Levy complexity at least $\Pi_n$ (for $n \ge 2$) and it is not an example of identity crisis, then it usually implies that any such cardinal is $\Sigma_n$-correct, and if there if an A-cardinal, where A is $\Sigma_n$-definable, then the least A-cardinal is less than the least $\Sigma_n$-correct cardinal; indeed, if $\lambda$ is a $\Sigma_n$-correct cardinal and there is an A-cardinal greater than or equal to $\lambda$, then $\lambda$ is a limit of A-cardinals, and if $\lambda$ is $\Sigma_{n+1}$-correct and there is an A-cardinal greater than or equal to $\lambda$, there are unboundedly many A-cardinals.

Thus the size order for non- identity crisis $\Pi_1$-, $\Delta_2$- or $\Sigma_2$-definable large cardinal properties is largely the same as the strength order:

• An otherwordly cardinal (also known as 0-extendible) is a limit of worldly cardinals.
• A wordly cardinal of uncountable cofinality is a limit of otherworldly cardinals.
• An inaccessible cardinal $\kappa$ is a limit of worldly cardinals of every cofinality less than $\kappa$ (indeed, a limit of a closed unbounded set of wordly cardinals).
• An inaccessible cardinal of non-trivial Carmody degree is a limit of inaccessible cardinals of all degrees less than its own.
• A $\Delta_2$-Mahlo (eqivalently $\Sigma_2$-Mahlo) cardinal $\kappa$ is a limit of inaccessible cardinals of every degree definable with parameters less than $\kappa$.
• If $n \ge 2$, a $\Pi_n$-Mahlo cardinal (equivalently $\Sigma_{n+1}$-Mahlo) is a limit of $\Sigma_n$-Mahlo cardinals.
• A $\Pi_\omega$-Mahlo cardinal is a limit of $\Pi_n$-Mahlo cardinals for every $n \lt \omega$
• A 0-pseudo-uplifting cardinal (that is, an inaccessible otherworldly cardinal) is a limit of $\Pi_\omega$-Mahlo cardinals.
• A 0-uplifting cardinal (that is, an inaccessible cardinal that is otherworldly to an inaccessible target) is a limit of 0-pseudo-uplifting cardinals.
• A Mahlo cardinal is a limit of 0-uplifting cardinals.
• A Mahlo cardinal $\kappa$ of non-trivial Mahlo degree $\beta \lt \kappa^+$ is a limit of Mahlo cardinals $\lambda$ of degree $g_{\gamma}(\lambda)$ for every $\gamma \lt \beta$, where $\langle g_{\gamma} | \gamma \lt \beta \rangle$ is a canonical sequence of functions.
• A weakly compact (equivalently, $\Pi^1_1$-indescribable) cardinal is a limit of greatly Mahlo cardinals (assuming I correctly understand that a cardinal is greatly Mahlo iff it is $\beta$-Mahlo for every $\beta \lt \kappa^+$).
• A $\Pi^1_n$-indescribable cardinal (equivalently $\Sigma^1_{n+1}$indescribable) is a limit of $\Pi^1_m$-indescribable cardinals for $1 \le m \lt n$
• If I understand correctly, a $\Pi^m_{n_0}$-indescribable cardinal is a limit of $\Pi^m_n$-indescribable cardinals for $0 \le n \lt n_0$, where $\Pi^{m+1}_0$-indescribable is equivalent to $\Pi^m_n$-indescribable for every $n \lt \omega$. A cardinal is said to be totally indescribable if it is $\Pi^m_n$-indescribable for all $m, n \lt \omega$.
• A $f(\kappa)$-strongly unfoldable cardinal $\kappa$ is a limit of $g(\lambda)$-strongly unfoldable if $f$ and $g$ are functions that are $\Delta_2$-definable with parameters in $V_\kappa$ and $g(\alpha) \lt f(\alpha)$ for almost all $\alpha \lt \kappa$. A cardinal is $+0$-strongly unfoldable iff it is weakly compact and $+n$-strongly unfoldable iff it is $\Pi^{n+1}_1$-indescribable (by Hauser's characterization of indescribable cardinals).
• A weakly superstrong cardinal $\kappa$ is $f(\kappa)$-strongly unfoldable for every function $f$ that is $\Delta_2$-definable with parameters in $V_\kappa$ and thus a limit of $g(\lambda)$-strongly unfoldable cardinals $\lambda$ for every such function $g$.
• A subtle cardinal is a limit of weakly superstrong cardinals.
• A weakly ineffable (=almost ineffable) cardinal is a limit of subtle cardinals.
• An ineffable cardinal is a limit of weakly ineffable cardinals.
• A 2-subtle cardinal (or 3-subtle depending on the how you define them) is a limit of ineffable cardinals. More generally, an $n$-weakly ineffable cardinal is a limit of $n$-subtle cardinals, an $n$-ineffable cardinal is a limit of $n$-weakly ineffable cardinals, and an $n+1$-subtle cardinal is a limit of $n$-ineffable cardinals. A cardinal that is $n$-ineffable for every $n \lt \omega$ is said to be totally ineffable.
• A completely ineffable cardinal is a limit of totally ineffable cardinals.
• A weakly Ramsey cardinal is a limit of completely ineffable cardinals.
• An $\alpha$-iterable cardinal (where $\alpha \le \omega_1$) is a limit of $\beta$-iterable cardinals for $1 \le \beta \lt \alpha$, where 1-iterable is the same as weakly Ramsey.
• The $\alpha$-Erdős cardinal is a limit of $\alpha$-iterable cardinals and the least $\alpha+1$-iterable cardinal is greater than the $\alpha$-Erdős cardinal if $\alpha$ is an additively indecomposable ordinal greater than or equal to $\omega_1$. I think that for every uncountable $\alpha$, the $\alpha$-Erdős cardinal is a limit of $\omega_1$-iterable cardinals.
• A cardinal $\kappa$ is almost Ramsey iff for every $\alpha \lt \kappa$, there is an $\alpha$-Erdős cardinal less than $\kappa$.
• A Ramsey cardinal is a limit of almost Ramsey cardinals.
• An ineffably Ramsey cardinal is a limit of Ramsey cardinals. More generally, if I understand correctly, a $\Pi_\alpha$-Ramsey cardinal is a limit of $g_\beta (\lambda)$-Ramsey cardinals $\lambda$ if $\langle g_{\beta} | \beta \lt \alpha \rangle$ is a canonical sequence of functions. A cardinal $\kappa$ that is $\Pi_\alpha$-Ramsey is said to be completely Ramsey.
• An almost fully Ramsey cardinal is a limit of completely Ramsey cardinals (if I remember correctly).
• A strongly Ramsey cardinal is a limit of almost fully Ramsey cardinals.
• A super Ramsey cardinal is a limit of strongly Ramsey cardinals.
• A fully Ramsey cardinal is a limit of super Ramsey cardinals.
• A locally measurable cardinal is a limit of fully Ramsey cardinals.
• A measurable cardinal is a limit of locally measurable cardinals.
• An $f(\kappa)$-strong cardinal $\kappa$ (where $f$ is a $\Delta_2$-definable function) is a limit of $g(\lambda)$-strong cardinals $\lambda$ for functions $g$ such that $g(\alpha) \lt f(\alpha)$ for almost all $\alpha \lt \kappa$. Measurable is equivalent to $+1$-strong.
• A Woodin cardinal is a limit of $f(\lambda)$-strong cardinals $\lambda$ for every function $f: \kappa \to \kappa$.
• A weakly hyper-Woodin cardinal is a limit of Woodin cardinals.
• A Shelah cardinal is a limit of weakly hyper-Woodin cardinals.
• A hyper-Woodin cardinal is a limit of Shelah cardinals.
• A superstrong cardinal is a limit of hyper-Woodin cardinals.
• A $+1$-extendible cardinal is a limit of superstrong cardinals.
• A subcompact cardinal is a limit of $+1$-extendible cardinals.
• A quasicompact cardinal is a limit of subcompact cardinals.
• A cardinal $\kappa$ that is $\beth_{\kappa+1}$-supercompact is a limit of quasicompact cardinals.
• A cardinal $\kappa$ that is 2-fold (or $n+1$-fold) $f(\kappa)$-strong is a limit of cardinals $\lambda$ that are (n-fold) $\beth_{f(\lambda)}$-supercompact if $f$ is a $\Delta_2$-definable function such that $f(\kappa)$ is a successor or a limit of cofinality greater than or equal to $\kappa$. A cardinal $\kappa$ that is (n-fold) $f(\kappa)+1$-extendible is a limit of cardinals $\lambda$ that are 2-fold ($n+1$-fold) $f(\lambda)$-strong. A cardinal $\kappa$ that is (n-fold) $\beth_{f(\kappa)}$-supercompact is a limit of cardinals $\lambda$ that are (n-fold) $f(\kappa)$-extendible if $f$ is a $\Delta_2$-definable function such that $f(\kappa)$ is a successor of a limit of cofinality greater than or equal to $\kappa$.
• A (n+fold) Vopěnka cardinal (equivalently [n-fold] Woodin for supercompactness, equivalently 2-fold [n+1-fold] Woodin) $\kappa$ is, for every function $f: \kappa \to \kappa$, a limit of cardinals $\lambda$ that are (n+fold) $f(\lambda)$-extendible, (n+fold) $\beth_{f(\lambda)}$-supercompact and 2-fold (n+1-fold) $f(\lambda)$-strong.
• A 2-fold (n+1-fold) Shelah cardinal (equivalently [n-fold] Shelah for supercompactness) is a limit of 2-fold (n+1-fold) Woodin cardinals.
• A (n+fold) high jump cardinal is a limit of (n+fold) Shelah for supercompactness cardinals.
• A (n-fold) almost huge cardinal is a limit of (n-fold) high jump cardinals.
• A 2-fold (n+1-fold) 0-extendible cardinal is a limit of (n-fold) almost huge cardinals.
• A (n-fold) huge cardinal is a limit of 2-fold (n+1-fold) 0-extendible cardnals.
• A 2-fold (n+1-fold) superstrong cardinal is a limit of huge cardinals.
• An I3 (also known as $E_0$) critical point is a limit of $\lt \omega$-huge cardinals (that is, cardinals that are n-huge for every n) and so is an I3 critical supremum.
• An $\omega$-fold Vopěnka cardinal is a limit of $E_0$ critical points and $E_0$ critical suprema
• An $IE_\omega$ critical point is a limit of $\omega$-fold Vopěnka cardinals. An $IE_{\alpha+\omega}$ critical point is a limit of $IE_{\alpha}$ critical points and suprema for countable $\alpha$. $IE_{\omega_1}$ is eqivalent to $IE$.
• An I2 (equivalently $E_1$) critical point (also known as an $\omega$-fold superstrong cardinal) is a limit of $IE$ critical points and suprema.
• An $\omega$-fold Woodin (=$W-E_1$) cardinal is a limit of I2 critical points and suprema. More generally, a $W-E_n$ cardinal is a limit of $E_n$ critical points and suprema.
• An $E_{n+1}$ critical point is a limit of $W-E_n$ cardinals.
• An I1 (=$E_\omega$) critical point is a limit of cardinals that are $E_n$ critical points for all $n \lt \omega$.
• An I0 critical point is a limit of I1 critical points and suprema.

As noted above, the least $\Sigma_2$-correct cardinal is greater than the least A-cardinal for every property A listed above. The following $\Pi_2$- or $\Sigma_3$-definable properties imply $\Sigma_2$-correctness:

• A $\Sigma_2$-reflecting (=$\Sigma_2$-correct and inaccessible) cardinal is a limit of $\Sigma_2$-correct cardinals and $\Sigma_2$-Mahlo cardinals.
• A strongly unfoldable cardinal is, for every $\Delta_2$-definable function $f$, a limit of $\Sigma_2$-reflecting and $f$-strongly unfoldable cardinals.
• A strong cardinal is, for every $\Delta_2$-definable function $f$, a limit of strongly unfoldable and $f$-strong cardinals.
• A $C^{(2)}$-superstrong cardinal is a limit of strong and superstrong cardinals.
• A supercompact cardinal is, for every $\Delta_2$-definable function $f$, a limit of $C^{(2)}$-superstrong and $f$-supercompact cardinals.
• An $f$-hypercompact cardinal (where $f$ is a $\Delta_2$-definable function) is a limit of $g$-hypercompact cardinals for every $\Delta_2$-definable $g$ such that $g(\alpha) \lt f(\alpha)$ for all $\alpha$. 1-hypercompact is the same as supercompact. A cardinal that is $\alpha$-hypercompact for every ordinal $\alpha$ is called hypercompact.
• An enhanced supercompact cardinal is a limit of hypercompact cardinals.
• A high-jump-with-unbounded-excess-closure cardinal is a limit of enhanced supercompact and high jump cardinals.
• A $C^{(2)}$-huge cardinal is a limit of high-jump-with-unbounded-excess-closure cardinals.

The least $\Sigma_3$-correct cardinal is greater than the least A-cardinal for every property A listed above. The following $\Pi_3$- or $\Sigma_4$-definable properties imply $\Sigma_3$-correctness:

• A totally otherworldly cardinal is a limit of $\Sigma_3$-correct and otherworldly cardinals.
• A $\Sigma_3$-reflecting (=$\Sigma_3$-correct and inaccessible) cardinal is a limit of totally otherworldly cardinals and $\Pi_2$-Mahlo cardinals.
• A pseudo-uplifting (=totally otherworldly and inacessible) cardinal is a limit of $\Sigma_3$-reflecting and $\Pi_\omega$-Mahlo cardinals.
• An uplifting cardinal (=inaccessible and totally otherworldly to unboundedly many inaccessible cardinals) is a limit of pseudo-uplifting cardinals.
• A superstrongly unfoldable (equivalently strongly uplifting) cardinal is a limit of uplifting, strongly unfoldable and weakly superstrong cardinals.
• A globally superstrong cardinal is a limit of superstrongly unfoldable, strong and superstrong cardinals.
• A $C^{(3)}$-superstrong cardinal is a limit of globally superstrong and $C^{(2)}$-superstrong cardinals.
• An extendible (equivalently 2-fold strong) cardinal is a limit of $C^{(3)}$-superstrong and hypercompact cardinals.
• A (n-fold) super high-jump cardinal is a limit of (n-fold) extendible and high-jump cardinals.
• A (n-fold) super almost huge cardinal is a limit of (n-fold) super high-jump, high-jump-with-unbounded-excess-closure and almost huge cardinals.
• A (n-fold) superhuge cardinal is a limit of (n-fold) super almost huge and huge cardinals.
• A $C^{(3)}$-huge cardinal is a limit of superhuge and $C^{(2)}$-huge cardinals.
• An (n-fold) ultrahuge cardinal is a limit of $C^{(3)}$-huge cardinals.
• A 2-fold (n-fold) globally superstrong cardinal is a limit of ultrahuge cardinals (at least I think so)
• A (n-fold) hyperhuge cardinal (also called 2-fold [n+1-fold] supercompact, equivalently 2-fold [n+1-fold] extendible, equivalently 3-fold [n+2-fold] strong) is, for $\Delta_2$-definable function $f$, a limit of 2-fold (n+1-fold) globally superstrong and 2-fold [n+1-fold] $f$-extendible cardinals.
• An $\omega$-fold extendible (=$P-E_0$) cardinal is a limit of $\lt \omega$-fold extendible cardinals that are $E_0$ critical points.
• An $\omega$-fold strong (=$P-E_1$) cardinal is a limit of $\omega$-fold extendible, $\omega$-fold superstrong and $\omega$-fold Woodin cardinals.
• More generally, a $P-E_{n+1}$ cardinal is a limit of $P-E_n$, $E_n$ and $W-E_n$ cardinals.

The least $\Sigma_4$-correct cardinal is greater than the least A-cardinal for every property A listed above. The following $\Pi_n$- or $\Sigma_{n+1}$-definable properties imply $\Sigma_n$-correctness (for large enough $n$):

• A $\Sigma_n$-reflecting (=$\Sigma_n$-correct and inaccessible) cardinal is a limit of $\Sigma_n$-correct cardinals and $\Pi_{n-1}$-Mahlo cardinals.
• A globally $C_{(n)}$-superstrong cardinal is a limit of $\Sigma_{n+2}$-reflecting and $C_{(n)}$-superstrong cardinals.
• A $C_{(n+2)}$-superstrong cardinal is a limit of globally $C_{(n)}$-superstrong and $C_{(n+1)}$-superstrong cardinals.
• A $C_{(n)}$-extendible cardinal is a limit of $C_{(n+2)}$-superstrong and (if $n \ge 2$) $C_{(n-1)}$-extendible cardinals, where $C_{(1)}$-extendible is the same as extendible.
• A $C_{(n)}$-superhuge cardinal is a limit of $C_{(n)}$-extendible, $C_{(n)}$-huge and (if $n \ge 2$) $C_{(n-1)}$-superhuge cardinals, where $C_{(1)}$-superhuge is the same as superhuge.
• A $C_{(n)}$-huge cardinal is a limit of $C_{(n)}$-superhuge and (n+1)-huge cardinals.

Examples of identity crisis include the following:

• As pointed out in Mohammad Golshani's answer, whether the least $\Sigma^m_n$-indescribable cardinal is greater or less than the least $\Pi^m_n$-indescribable cardinal, for $2 \le m$ and $1 \le n$, is independent. I think $\Sigma^m_{n+1}$-indescribable cardinals are always limits of $\Pi^m_n$-indescribable cardinals and $\Pi^m_{n+1}$-indescribable cardinals are always limits of $\Sigma^m_n$-indescribable cardinals.
• An unfoldable cardinal is weakly compact and a strongly unfoldable cardinal is unfoldable. In the constructible universe every unfoldable cardinal is strongly unfoldable but $\omega_1$-iterable cardinals are also unfoldable and Hamkins metioned that it is consistent that the least weakly compact cardinal is unfoldable.
• A tall cardinal is measurable and a strong cardinal is tall. It is consistent that the least tall cardinal is strong or the least measurable cardinal is tall. Tall cardinals are equiconsistent with strong cardinals.
• A strongly compact cardinal is tall, and thus measurable, and a supercompact cardinal is strongly compact. It is consistent that the least strongly compact cardinal is supercompact or the least measurable cardinal is strongly compact.