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?

  • $\begingroup$ I'm sure what you are asking here. Are you trying to write a formula with parameters to describe Mahlo and hyper-Mahlo, and so on? $\endgroup$ – Asaf Karagila Sep 5 '11 at 6:45
  • $\begingroup$ @ Asaf : hope my update clarifies the question. $\endgroup$ – Ewan Delanoy Sep 5 '11 at 15:34
  • $\begingroup$ I undid some of the spaces to ease readability. $\endgroup$ – Asaf Karagila Sep 5 '11 at 15:41
  • $\begingroup$ Ewan, it is still unclear. Are you trying to write a first order definition (or a schema of definitions) which will capture exactly all the cardinals which are $\alpha$-Mahlo for some $\alpha$? $\endgroup$ – Asaf Karagila Sep 5 '11 at 15:44
  • $\begingroup$ Asaf: I think he's after the concept of 'Mahloness'. The problem, of course, is the one that is noted in the post itself; given that the existence of Mahlo cardinals themselves can't be deduced from ZFC, there's no way of concluding that the existence of any of these cardinals is consistent with ZFC. More broadly, he seems to be asking about the nature of large cardinal hypotheses in general and it sounds like some of Woodin's results about their ranking would be of interest here... $\endgroup$ – Steven Stadnicki Sep 5 '11 at 17:39

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.


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