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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'.

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2 Answers 2

The numbering system Godel used to prove incompleteness did rely on properties of primes. To use a numbering system to prove incompleteness one must be able to distinguish (numbers that encode) terms from (numbers that encode) strings of terms; strings of terms from well-formed formulas; formulas from finite sequences of formulas; sequences of formulas that are proofs of other formulas; etc. There is no reason, other than convenience, for a numbering system to rely on properties of primes; there is no difficulty in constructing a numbering system, suitable for proving incompleteness, without relying on properties of primes.

In effect, Godel's result has nothing to do with Euclid's.

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I'm not an expert in this field, but here are some thoughts I had while reading your post.

Godel worked in the axiom system of Peano arithmetic which is strong enough to define prime numbers and prove the infinitude of primes. His proof takes this fact (fact within Peano arithmetic!) and uses it to prove the incompleteness theorem.

You said, "Roughly, Godel's incompleteness theorems proves a formal, system of mathematics is unable to prove it's own consistency."

That's almost correct. He didn't show ALL formal systems can't prove their own consistency, only ones that are sufficiently strong for his proof to apply (so being able to define and prove the infinitude of primes would of course be important).

"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?"

Not sure how you lept to this. Just because a system can't prove its own consistency doesn't mean a theorem that is derived from its axioms is somehow not true anymore. Anything derived by the axioms is by definition true, one thing derived being the infinitude of primes. Primeness as a property is a definition: you say "A number is prime if and only if...", the property of "primeness" should be tautological since primeness is a definition.

"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?"

I'm not sure how to answer this question, but this link may be interesting to you: http://mathoverflow.net/questions/75995/formalizing-euclids-proof-of-the-infinitude-of-primes

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