The existence of concatenation functions in Godel Numbering? I know that there are many schema of Gödel Numbering, and each has its own method of Concatenation, n★m. 
But is there a general proof that shows 'For every Gödel Numbering scheme there exists a Concatenation Function'?
it is more the other way around without contentation no Godel numbering.

Godelnumbering is just a translation from formulas and proofs to numbers and from the Godelnumbers back to the same  formulas and proofs. 

I think in principle any translation system that can do this can be used as godelnumbering system.

Be aware:


The numbering system  also needs to be able to translate proofs to numbers and back
all translations need to be lossless and exact
it must all be done in the axioms of arithmetic.


So it is not only translating a formula to an number , it is much more

Hope this helps
it is more the other way around without contentation no Godel numbering.

Godelnumbering is just a translation from formulas and proofs to numbers and from the Godelnumbers back to the same  formulas and proofs. 

I think in principle any translation system that can do this can be used as godelnumbering system.

Be aware:


The numbering system  also needs to be able to translate proofs to numbers and back
all translations need to be lossless and exact
it must all be done in the axioms of arithmetic.


So it is not only translating a formula to an number , it is much more

Hope this helps