# Numeralwise expressibility in logic

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!

• It must be read more or less literally : we know "facts" about natural numbers. Are the expressive and deductive resources of first-order arithmetic "strong enough" to formalize and prove them ? On the deductive side, the answer is NO (see G's Incompleteness Th). – Mauro ALLEGRANZA Oct 10 '18 at 6:00
• @MauroALLEGRANZA Aren't the facts we know about natural numbers coming from proofs about them? Is there some fact that does not come from a proof? – Daniels Krimans Oct 10 '18 at 6:06
• Yes, they comes form proofs... but not necessarily from proofs in FOL arithmetic. Humans learned to count well before math logic. – Mauro ALLEGRANZA Oct 10 '18 at 6:09
• @MauroALLEGRANZA does it mean that truth here means relative truth with respect to some different formal system? – Daniels Krimans Oct 10 '18 at 6:17
• You can see Introduction to Numeralwise Expressibility and see Representability. – Mauro ALLEGRANZA Oct 10 '18 at 12:34