User blog comment:Ytosk/Trying to define Bowers' K(n) systems/@comment-35470197-20191031094256/@comment-35470197-20200206234838

> I don't see why it shouldn't be. Set membership is a relation in set theory, just like r is in that axiomatic system.

Set theory is built under the formal language which includes only ∈ as a non-equality relation symbol. On the other hand, you are not restricting your "system" to characterise a single relation symbol r.

> let's define Def(x) to be the definability of x,

As I said, you are not defining the notion of definability, which is ambiguous in this context unlike usual mathematics. Please write down the definition of Def(x) using mathematical formulae.

> the formula used to define x can use the set membership relation.

I am afraid that you are comfounding ∈ on V and a relation symbol "∈" in a formal language.

> MK(n,m)=max({|L| | א‎0>|L|∧∃o∃S∃r((o is a K(m) system)∧(r well-orders S)∧∀x∈S((the axioms of o can prove the existence of x)∧(Def(S)+Def(r)+Def(x)≤n⇒∀y∈S(r(y,x)⇒y∈L)))∧∀y∈L∃x∈S((the axioms of o can prove the existence of x)∧Def(S)+Def(r)+Def(x)≤n∧r(y,x)))})

The predicate "the axioms of o can prove the existence of x" is ill-defined. Please clarify the definition using mathematical formulae. I recommend you to avoid to use natural language in the definition, because they are causing serious issues.