# On $ORD$ “satisfying” large cardinal properties

It is common for people to reference that "$ORD$ is strongly inaccessible" or "$ORD$ is Mahlo". The first of these (when formalized) is easily proven in ZFC. However, the latter is equiconsistent to the existence of Reflecting cardinals. My question is (informally) how many large cardinal properties one could give $ORD$.

Generally, $ORD$ having a large cardinal property is somewhat weaker than said large cardinal property. Generally, if $\kappa$ has the property then $V_\kappa$ or $V_{\kappa+1}$ satisfies that $ORD$ has that property.

First, I question $ORD$'s weak compactness. Let $\mathcal{L}_{\infty,\infty}$ be $\bigcup_{\alpha\in ORD}\mathcal{L}_{\alpha,\alpha}$. Let $ORD$ be weakly compact iff for any $\mathcal{L}_{\infty,\infty}$-theory $T$ (allowing $T$ to be a proper class), $Con(T)$ iff for every set $t\subseteq T$, $Con(t)$.

Secondly, I question $ORD$'s indescribability. Let $ORD$ be $Q$-indescribable iff:

For any $Q$-sentence $\phi$ in the language of $\{\in,P\}$ for $P$ an unary predicate symbol and any class $A$:

$$\langle V;\in,A\rangle\models\phi\Leftrightarrow\exists\alpha(\langle V_\alpha;\in,A\cap V_\alpha\rangle\models\phi)$$

Let $ORD$ be totally indescribable iff it is $\Pi_m^n$-indescribable for every $m$ and $n$.

Thirdly, I question $ORD$'s subtlety (I skipped a few because I could not find a way to translate them to $ORD$; namely strongly uplifting and unfoldable cardinals). Let $ORD$ be subtle iff:

For any sequence $\langle A_\alpha:\alpha\in ORD\rangle$ such that for any $\alpha$, $A_\alpha$ is a set of ordinals and any club class of ordinals $C$, there are $\alpha<\beta$ in $C$ such that $A_\beta\cap\alpha=A_\alpha$.

Next, I question $ORD$'s $n$-ineffability. For a class $W$, let $[W]^n$ be the class of all subsets of $W$ with cardinality $n$. Let $ORD$ be $n$-ineffable iff:

For every class function $f:[ORD]^n\rightarrow 2$, there is a class of ordinals $H$ which is stationary in $ORD$ where $f(x)=f(y)$ for any two $x$ and $y$ in $[H]^n$.

My main questions are whether or not these properties have the expected consistency strength; that is "$ORD$ is Mahlo" $\leq$ "$ORD$ is weakly compact" $\leq$ "$ORD$ is totally indescribable" $\leq$ "$ORD$ is subtle"... and whether or not "$ORD$ is $A$" $<$ $\exists\kappa(\kappa\;\mathrm{is}\;A)$.

THE PERSON ANSWERING THIS QUESTION DOES NOT HAVE TO ANSWER THE CONSISTENCY STRENGTH OF EVERY ONE. I will accept any answer as long as it details into at least one of the stated properties here.

Thank you!

EDIT: After doing some research, I have come to the conclusion that for any $\kappa$, $V_\kappa\models$ "$ORD$ is $Q$-indescribable" iff $\kappa$ is $Q$-indescribable. Thus, $Q$-indescribability of a cardinal is strictly stronger than $Q$-indescribability of $ORD$.

• Joel's answer in the following related question might be of your interest: How strong are large cardinal properties of Ord? – Morteza Azad Oct 31 '17 at 22:48
• While that was a very useful argument, it did not provide answers to these questions. As Asaf said, the definition for "$ORD$ satisfies $P$" is very different from the intuition. Since I went for an intuition-based approach, I don't believe this applies. Thank you for suggesting that though, it was an interesting read. – Keith Millar Nov 1 '17 at 0:21
• It is unclear to me what you are asking. The way you formulate "ORD is weakly compact" is not a sentence in ZFC but rather a scheme of sentences. Please specify what your base theory is.(Probably a theory in which you can talk explicitly about classes?) – Goldstern Mar 13 '18 at 23:14