I am reading Chapter 8 Section 41 of Kleene's "Introduction to metamathematics" and I have encountered notion of "numeralwise expressibility". Next is a quote from the textbook:
"Let $P(x_1, ..., x_n)$ be an intuitive number-theoretic predicate. We say that $P(x_1, ... , x_n)$ is numeralwise expressible in the formal system, if there is a formula $\mathrm{P(x_1,...,x_n)}$ with no free variables other than the distinct variables $\mathrm{x_1,...,x_n}$ such that, for each particular $n$-tuple of natural numbers $x_1, ... , x_n$, (i) if $P(x_1, ..., x_n)$ is true then $\vdash \mathrm{P(x_1, ..., x_n)}$ and (ii) if $P(x_1, ..., x_n)$ is false then $\vdash \lnot \mathrm{P(x_1, ..., x_n)}$. "
Question 1: I have trouble understanding what does it mean for something to be true or false without any reference to deducibility. I thought that the whole point of formalism is that we introduce finitary concept of deducibility to encode "truth" as being "provability". I think there might be some misconception on my side here.
Question 2: What is the use of such definition? How is it helpful? Why do we need it?
Question 3: How do I check whether intuitive number-theoretic predicate is true or false?
I would appreciate any suggestions or comments!