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 $.

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?