When Gödel numbered statements, for instance modus ponens and connectives got their own numbers, does it matter which number each connective gets as long as they are different?
Sometimes I'm not sure if a statement is ¬($~A\to B$) or $~A\to ¬B$.
What is the Gödel number for ¬($~A\to B$) ("not modus ponens") e.g. saying for example "just because we change the interest rate doesn't mean that the trade deficit will narrow", or is it arbitrary as long as it is unique?
I read in a paper that implication has number 13. Is implication always number 13, regardless of which proof or formula we are working with?