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?
If $T,H$ are ultraconsistent then $T \cup H$ is ultraconsistent.
If $T$ is ultraconsistent and $G$ is a fragment of $T$ then $G$ is ultraconsistent.
Can we have a theory that is ultraconsistent and yet false?