Is primeness a predicative property? Earlier this year, I jotted down some thoughts in a paper whether Euclid's proof of infinitude of prime numbers is tautological arguing that prime numbers are prime because of virtue of primeness.
Before you consider the following points, a caveat to bear in mind would be that I am not an expert in logic or meta-mathematics. [Background: I am an undergrad in college and I am familiar with linear algebra and multivariate calculus but tried to penetrate model or proof theoretic logic on my own in vain.]
- If I understand correctly, Godel used a numbering or encoding to "assigns to each symbol and well-formed formula of some formal language a unique natural number" (Godel numbering)
- Godel numbering relies on prime factorization.
- In order to utilize prime factorization, primeness must be defined (?).
- Roughly, Godel's incompleteness theorems proves a formal, system of mathematics is unable to prove it's own consistency.
Are we then back in full circle? Then this begs the question if Godel utilized the concept of primeness to establish his result, in order to determine if Euclid's proof of infinitude of prime number is tautological or circular, how can we use Godel's results granted the fact he assumed(?) the concept of primeness in the first place?
Hence, my original question, is the concept of primeness impredicative (bearing in mind there is no accepted formal,rigorous definition of predicativity) ?
Secondly, suppose I wanted to investigate meta-mathematical properties of Euclid's proof. If I convert the proof to predicate-logic, what apparatus or theorems will I use to find out if indeed the proof is circular or not?
(p.s. I did check this question)
Edit: Here is the formal proof of infinitude of primes in metamath website. Loosely, I was asking how to go 'beyond' or 'outside' this system to ensure it's 'validity' or 'soundness'.