User blog comment:Boboris02/MBOT/@comment-1605058-20161230135452/@comment-1605058-20161230151740

"I believe theorems can exist independently of axioms" Your belief is wrong. No formal deductive system can have theorems if you require all of them need to be proven from some other theorems. However, the part "Some theorems are set as default" sounds precisely like axioms - theorems which you don't require a proof for.

So your \(\Delta_\kappa\) is what nearly everyone else in the world would write as \(\Delta(\kappa)\)? Glad to have that clarified.

I am probably misunderstanding, but with that definition everything is a function. For let's take, say, the letter F. Then there exists a \(\rho\) which is equal to F, namely F itself. So F is a valid \(\Delta_\kappa\).

At the very least, if you want it to really mean a function, you have to mention dependence on \(\kappa\) (so you can say something along the lines "for every \(\kappa\) there is a \(\rho\) such that \(\Delta_\kappa=\rho\)"). But even then, as soon as you specify that \(\Delta_\kappa\) is an object depending on \(\kappa\), you already have a function. No need to specify that this is equal to something - it is equal to itself.