$Define\space recursively:$  

$T_0 = T$
 
$T_{i+1} = T_i + Con(T_i) \ \  for \ every \ recursive \ ordinal \ i$
 
$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? $