The existence of concatenation functions in Godel Numbering?

Question

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'?

Answer 1

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