Define recursively:  

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

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

Say that $T$ is *ultraconsistent* iff $$\forall i \, ( i\text{ is  a recursive  ordinal }\implies \operatorname{Con}(T_i)),$$  where  a recursive  ordinal  is  an  ordinal  below the Church-Kleene ordinal $\omega _{1}^{\mathrm {CK} } $.

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 that is ultraconsistent and yet false?