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Is there an axiom scheme exhausting all types of Mahlo cardinals?

Mahlo cardinals may be considered as the first stage in the following construction : let $C_{0,0}$ be the class of all inacessible cardinals. Then define by transfinite induction $\beta \in C_{0,\alpha+1}$ iff any normal function $\beta \to \beta$ has a fixed point which is in $C_{0,\alpha}$, and $C_{0,\lambda}=\cup_{\mu < \lambda} C_{0,\mu}$ for limit $\lambda$.

That leads to axiom $A_0$ : $C_{0,\alpha}$ is nonempty for any ordinal $\alpha$.

This axiom does not exhaust all possible Mahlo cardinals, however, because we may diagonalize and define $C_{1,0}$ as the class of all cardinals $\beta$ such that $\beta \in C_{0,\gamma}$ for all $\gamma < \beta$. Then we induct transfinitely again: $\beta \in C_{1,\alpha+1}$ iff any normal function $\beta \to \beta$ has a fixed point which is in $C_{1,\alpha}$, and $C_{1,\lambda}=\cup_{\mu < \lambda} C_{1,\mu}$ for limit $\lambda$.

That leads to axiom $A_1$ : $C_{1,\alpha}$ is nonempty for any ordinal $\alpha$.

But this does not exhaust all possible Mahlo cardinals yet, because one may still diagonalize and define classes $C_{2,\alpha}$, $C_{3,\alpha}$ etc. and in fact we may define $C_{\beta,\alpha}$ for any ordinals $\beta$ and $\alpha$, etc.

One feels that it is impossible to formulate an axiom that would exhaust all this hierarchy. Is that impression formalized in some theorem?

UPDATE (09/05/2011) : formally what I mean is this : let $\phi$ is an axiom or axiom scheme of set theory (so that $\phi$ is a meaningful sequence of quantifiers $\forall,\exists$, of logical connectives $\Rightarrow, \vee, \wedge, \rceil$, of any number of variables $x_1, \ldots x_n$, any number of formula variables $\phi_1, \phi_2, \ldots ,\phi_m$ (when we have an axiom scheme) and the $\in$ symbol). Then my guess is that $\phi$ does not suffice as an axiom to exhaust all type of Mahlo cardinals ; that there will always be some " Mahlo class" $C$ of cardinals such that the non-emptiness of $C$ cannot be deduced from $\phi$. In fact, if $\kappa$ is the smallest cardinal in $C$ then $(V_{\kappa},\in)$ is a model of $ZFC+\phi+$ " $C$ is empty". Of course, we are assuming that set theory is consistent and excluding the uninteresting case where $\phi$ is not consistent with $ZFC$. And my question is : is that guess of mine correct? Where does it appear in the literature?

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    @Steven: I only now see your comment, you can use an @-symbol to notify the user you are replying his comment (for next time...). As for ranking Mahlo cardinals, it is expressible in ZFC when a cardinal is Mahlo, it just unprovable if any cardinals are satisfying the formula (setting aside the issue of weakly inaccessible being possibly inconsistent with ZFC). I would imagine that there is a formula with an ordinal parameter such that $\varphi(\kappa,\alpha)$ if and only if $\kappa$ is $\alpha$-Mahlo.2011-09-13

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One can define, in the language of ZFC, the property you defined as $\kappa\in C_{\alpha,\beta}$. The main point is that, although $\alpha$ and $\beta$ seem to range over arbitrary ordinals (which can make a nested recursion difficult to formalize), you can restrict attention here to $\alpha,\beta\leq\kappa$, because $\kappa$ can't be any more Mahlo than that anyway. Then, for each $\kappa$, you're doing a perfectly ordinary nested recursion over pairs of ordinals below $\kappa$.

As a consequence, you can formulate, in the language of ZFC, an axiom saying that, for every $\alpha$ and $\beta$, there is at least one $\kappa$ (or even a proper class of $\kappa$'s if you're feeling generous) in $C_{\alpha,\beta}$.

For even stronger Mahlo-style axioms, you might look into the notion of "greatly Mahlo" cardinals.