I am reading a translation of Gödel's original paper about in completeness theorem and there are a couple things i don't understand.
Here is the document i am using primarily : http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf
I am just curious : In his system P, there is no symbol for addition/multiplication or any arithmetical formula, yet it is often mentionned "number-theoretical functions" or "number-theoretical formula", i guess they are "represented" by their extension type-n variables and signs.
But I guess there may not systematically be a formula for any classes of subsets of N representing a function/functionnal relation. Yet there is no symbols for arithmetic, then what about those "number-theoretical functions" or "number-theoretical formula" ?
Thank you
The signs of system $P$ include $0$ and $\mathit{succ}$ (see the beginning of section 2.1 in your reference). These together with the logical apparatus in the system are enough to define any primitive recursive relation (see section 2.5).