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An application of Gödel's incompleteness theorem?

  1. Jun 20, 2012 #1
    I have a problem in my logic course which I can't get my head around:

    I have to show that there is a well formed formula [itex]\mathcal{A}(x_1)[/itex] in the formal first order system for arithmetics, [itex]\mathcal{N}[/itex], with precisely one free variable [itex]x_1[/itex], such that [itex]\mathcal{A}(0^{(n)})[/itex] is a theorem in [itex]\mathcal{N}[/itex] for all [itex]n\in D_N[/itex], but where [itex]\forall x_1\mathcal{A}(x_1)[/itex] is not a theorem in [itex]\mathcal{N}[/itex]. Here [itex]D_N[/itex] denotes the set of natural numbers.

    My initial idea was to use the statement and proof of Gödel incompleteness theorem, but I get stuck in a bit of a circle argument with the ω-consistency, so perhaps my idea of using this theorem is all wrong.


    Edit: If it isn't obvious from the context, it is fair to assume that [itex]\mathcal{N}[/itex] is consistent.
     
  2. jcsd
  3. Jun 20, 2012 #2
    The question is just asking you to show that [itex]\mathcal{N}[/itex] is ω-incomplete, which Godel's theorem easily allows you to do, because the Godel statement is precisely a statement of the form [itex]\forall x_1\mathcal{A}(x_1)[/itex] where [itex]\mathcal{A}(0^{(n)})[/itex] is a theorem in [itex]\mathcal{N}[/itex] for all [itex]n\in D_N[/itex]. Because the Godel statement G can be informally stated as "For all [itex]x_1[/itex], G cannot be proved in [itex]\mathcal{N}[/itex] with an [itex]x_1[/itex]-lines-long proof." However, for each specific [itex]n\in D_N[/itex], there are only finitely many proofs that are n lines long, so just by writing down all the valid proofs in [itex]\mathcal{N}[/itex] that are n lines long, you can establish that there is no proof of G that is n lines long. (Or alternatively, you can state G as "For all [itex]x_1[/itex], G cannot be proved in [itex]\mathcal{N}[/itex] with the (valid) proof whose Godel number is [itex]x_1[/itex]", and then for each [itex]n\in D_N[/itex] you can show that the proof with Godel number n does not prove G.) Does that make sense?
     
    Last edited: Jun 20, 2012
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