$Define\space recursively:$

$T_0 = T$

$T_{i+1} = T_i + Con(T_i) \ \ for \ every \ recursive \ ordinal \ i$

$T_j = U\{T_i| i<j\} \ \ for \ every \ limit \ recursive \ ordinal \ j $

$Define: \ T \ is \ Ultraconsistent \iff \forall i \ ( i \ is \ a \ recursive \ ordinal \implies Con(T_i))$

$where \ a \ recursive \ ordinal \ is \ an \ ordinal \ below \ Church-Kleene \ ordinal$ ${\displaystyle \omega _{1}^{\mathrm {CK} }} $

$Does\space these\space statements\space follow?$

$1. \ T,H\ are \ Ultraconsistent \implies T \cup H \ is \ Ultraconsistent?$

$2. \ T \ is \ Ultraconsistent \wedge G \ is \ a \ fragment \ of \ T \implies G \ is \ Ultraconsistent?$

$ 3. \ Can \ we \ have \ a \ theory \ that \ is \ Ultraconsistent \ and \ yet \ FALSE? $