User blog comment:Ytosk/Trying to define Bowers' K(n) systems/@comment-35470197-20191031094256/@comment-35470197-20191101081850

> Now imagine a completely different axiomatic system. It has a binary relation r(a,b),

Do you understand that r is not canonically determined by the system? Since you are referring to "all" K(n) systems, such justification does not work.

As I said, you need to specify the way to make a system to include arithmetic. The existence of a way does not solve any problem.

For example, there are infinitely many ways to make set theory to include arithmetic, but we only refer to the standard model when we talk about natural numbers unless we specify another.

> unless you just flexed by using unnecessarily complex words.

Which word? I guess that you know Rayo's function and busy beaver function. It should not be "definability", because you are using it by yourself. Is it "first order", because you have not referring to the order of formal theories which you are considerring? Or are you comfounding syntax with semantics? If you do not understand what others say, it is good to point out it instead of doubting them.

> Is truth predicate really just something that outputs its input?

No, it is not kind of an identity. It is a (non-unique) predicate which evaluates whether a given formula is true or not.