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I remember reading something about a large cardinal axiom saying something like

If some cardinal $\kappa$ has some property $P$, then there should be a proper class of cardinals with the property $P$.

Of course this is inconsistent if you allow any property $P$, but this was used to justify for example the existence of uncountable inaccessible cardinals, because $\aleph_0$ is inaccessible, so there should be a proper class of inaccessible cardinals (because having only one inaccessible cardinal is not very homogeneous, and the ordinal hierarchy should be homogeneous).

Unfortunately I don't remember where I read that.

My questions are: Is there a precise formalisation of this axiom, and (if there is one) how high is it in the large cardinal hierarchy?

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There is such a statement in Penelope Maddy's Believing the Axioms (JSL 53, 1988). However, she only states it in an informal way. – François G. Dorais Nov 8 '11 at 20:18
Perhaps Foreman's potent axioms have somewhat similar properties: Axiom of resemblance: For all finite sequences of uncountable regular cardinals $\lambda_1<\cdots<\lambda_n$ there is a generic elementary embedding $j\:V\to M$ such that $j(\aleph_i)=\lambda_i$j for $1\le i\le n$ and ${\scr P}(\lambda_n)^V\subseteq M$. Potent axioms, TAMS 294(1986), 1-28. – Péter Komjáth Nov 8 '11 at 21:26
The book by Frank Drake (Set theory: an introduction to large cardinals) presents large cardinals along these lines. – Andrés Caicedo Nov 9 '11 at 7:26
@François yes, it's probably in Penelope Maddy's article that I read that. – Guillaume Brunerie Nov 10 '11 at 13:53
up vote 12 down vote accepted

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

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The MP axiom is equivalent to the following assertion of resurrectibility: any true statement $\varphi$ is forceable over every forcing extension. Thus, any truth that happens to be destroyed in one forcing extension, can be recovered by further forcing. This is itself an instance of the slogan, "anything that can happen once happens unboundedly often". – Joel David Hamkins Nov 9 '11 at 0:21
Hi Joel, in your statement of the strong reflection axioms, do we restrict the $\phi$ we consider to $\delta$-free sentences? Otherwise, won't we get some sort of problem with there existing a definable truth predicate for this theory? – Amit Kumar Gupta Nov 9 '11 at 21:16
Yes, that is what I meant. We only assert elementarity for formulas in the language of set theory, without $\delta$. – Joel David Hamkins Nov 9 '11 at 21:19

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