Here are two contexts in which such conclusions follow.
Strong reflection axioms. Consider the strong reflection axiom, sometimes denoted $V_\delta\prec V$, which is axiomatized in the language of set theory augmented with a constant symbol for $\delta$, axiomatized by the assertions $$\forall x\in V_\delta\ \ [\varphi(x)\iff \varphi^{V_\delta}(x)].$$
This theory is a conservative extension of ZFC, since every finite subset of the theory can be interpreted in any given model of ZFC by the reflection theorem. Meanwhile, in this theory, if the cardinal $\delta$ (or any larger cardinal) has a property $P$ expressible in the language of set theory, then $V$ satisfies $\exists\kappa P(\kappa)$, and so $V_\delta$ also satisfies this assertion, so there is a $\kappa\lt\delta$ with $P(\kappa)$. Similarly, if the collection of cardinals $\kappa$ with $P(\kappa)$ is bounded, then $V_\delta$ would have to agree on the bound by elementarity, but it cannot agree on the bound since $\delta$ itself has the property. So the collection of cardinals with property $P$ must be unbounded in the ordinals.
A stronger formulation of the strong reflection axiom includes the assertion that $\delta$ itself is inaccessible (or more), in which case it carries some large cardinal strength. It is exactly equiconsistent with the assertion "ORD is Mahlo", meaning the scheme asserting that every definable stationary proper class contains a regular cardinal, which is weaker in consistency strength than a single Mahlo cardinal.
Another seemingly stronger formulation of the theory, but still conservative over ZFC, is due to Feferman, and this asserts that there is a closed unbounded proper class $C$ of cardinals $\delta$, all with $V_\delta\prec V$. This theory can be stated as a scheme in the language of set theory augmented by a predicate symbol for $C$. Feferman proposed it as a suitable substitute and improvement of the use of universes in category theory, because it provides a graded hierarchy of universe concepts, which moreover agree between them and with the full universe on first-order set-theoretic truth.
The maximality principle. The maximality principle is the scheme asserting that any statement $\varphi$ which is forceable in such a way that it continues to be true in all further forcing extensions, is already true. This axiom is the main focus of my article, "A simple maximality principle", JSL 62, 2003, and was introduced independently by Stavi and Vaananen. The axiom asserts in short, that anything that you could make permanently true in forcing extensions, is already permanently true. The point now is that under MP, one gets your phenomenon:
Theorem. Under MP, if there is any inaccessible cardinal, then there is a proper class of such cardinals. And the same for Mahlo cardinals and many other large cardinal concepts.
Proof. Assume MP. If there are no inaccessible cardinals above some ordinal $\theta$, then consider the forcing to collapse $\theta$ to be countable. In the resulting forcing extension $V[G]$, there will be no inaccessible cardinals at all, and there never will be such cardinals in any further extension (because any inaccessible cardinal of $V[G][H]$ would also be inaccessible in $V[G]$). Thus, in $V$ the assertion that there are no inaccessible cardinals is forceably necessary, and so by MP, it must already be true. Thus, under MP, either there are no inaccessible cardinals or there are a proper class of them. The same argument works with Mahlo cardinals or any large cardinal concept that is downwards absolute. QED
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.
Best Answer
$\kappa$ is superhuge if for any $\gamma$, there exists $j: V\to M$ such that $$crit(j)=\kappa,$$ $$\gamma<j(\kappa)$$ and $${}^{j(\kappa)}M\subset M.$$ But $j(\kappa)$ (inaccessible in M) must be inaccessible in $V$ as well.