I want to know if there are any papers or research along the following lines. From a philosophical point of view there is no reason to think size has a limit, if we can describe it it exists, provided it does not contradict the axioms of our favourite set theory.
I think this can be expressed as a meta-axiom schema as follows. Let $AX$ be the set of axioms you like in the language $<\in >$ of set theory.
Axiom X - Let $\varphi $ be a statement with one free variable $\kappa$. If $AX\cup \{Cardinal[\kappa ], \varphi\}$ is consistent then $\exists\kappa\varphi$ is an axiom.$\square$
I would like to know if this could be converted to an internal axiom, or axiom schema, of set theory using extenders or such like. This would result in a non-conservative but consistent set theory in which all large cardinals exist.