User blog comment:Boboris02/MBOT/@comment-1605058-20161230135452/@comment-30118230-20161230141916

First,I believe theorems can exist independently of axioms.Some theorems are set as default for an existing case.

Second,\(\Delta_\kappa\) is the function (\(\Delta\)) and the input (\(\kappa\)).

Third,that means that there exists a Phi-system which would give precicely the same output for precicely the same input.

Also,yes! That's what it means.And that means that \(\rho\) exists if \(\Delta_\kappa = \rho\).\(\rho\) is the output and it only exists if there is a function,which would give that exact output for an input \(\kappa\).If a function does not give an output for any input,then it's not a function       (by MBOT's definition atleast).

EDIT:I explained this in the blog post.