Let $D$ be an effectively generated set of recursive ordinals that contains $0$ and that is closed under an effectively defined ordinal successor function "$+1$" and that have a well defined relation of ordinal strict smaller than relation $<$ on it, such that no two distinct elements of $D$ have the same order type. We'll simply label such a set as an *"effective set of recursive ordinals"*

Define recursively 

 - $T_0 = T$,
 - $T_{i+1} = T_i + \operatorname{Con}(T_i)$ for every $i \in D$,
 - $T_j = \bigcup\{T_i\mid i<j\}$ for every limit ordinal $j \in D $.

Where $T$ is an extension of $PA$.

Say that $T$ is $D$_*ultraconsistent* iff $\forall i \in D \ ( \operatorname{Con}(T_i)) $ 

Say that $T$ is *ultraconsistent* iff 

$\forall D (D$ is an *effective set of recursive ordinals*$ \implies T $ is  $D$_*ultraconsistent*$)$

Do the following statements follow? 

 1. If $T,H$ are ultraconsistent then $T \cup H$   is ultraconsistent. 

 2. If $T$  is ultraconsistent and $G$ is a fragment of $T$ then $G$ is  ultraconsistent.

 3. Can we have a theory $T$ that is ultraconsistent and yet $FALSE$?