• Pippen
    80
    I always wondered how to describe Gödel's Incompleteness Theorems in a nutshell, without all the technicalities, but still very close to what its idea is. I am interested to read if my description nails the idea of the Incompleteness Theorems or where I commit serious errors or confusions.

    p.s. I did this a while ago already, but I think my new version is better than the old one. Please only answer if you somehow familiar with the theorems!

    First Incompleteness Theorem

    We assume a consistent formal system S where we can syntactically correct formulate the following statement G: G <-> ~Proof(G). There are two cases within S:

    (1) G is provable, but then G is not provable (~Proof(G)) which is a contradiction and therefore impossible,

    (2) ~G is provable which means (~G & Proof(G)) v (G & ~Proof(G)), but that means in either case G will be proven which is a contradiction and therefore impossible.

    So our (consistent) system S cannot prove G or ~G and is therefore incomplete (or it could prove G or ~G if it was inconsistent for trivial reasons). Within S we can't decide if G or ~G is true, but of course from a meta-view we know that G is true. Gödel's "only" accomplishment was to show that G can be formulated syntactically correct in a special S called PM and therefore "infects" whole math (and yes, that was genius).

    Second Incompleteness Theorem

    Let's assume our system S again, this time strong enough to prove its First Incompleteness Theorem, i.e. if S is consistent then G which says G <-> ~Proof(G). Let's assume we could prove the consistency of S in S. Then by mp we could prove in S that G which is impossible due to the First Incompleteness Theorem, therefore the assumption must be false.
  • tim wood
    9.3k
    Details aside, as you say, this seems to me pretty good. Small point - or maybe not a small point. Godel had to distinguish between "proves" as in 1) "X proves Y," and "provability," as in 2) "X is provable."

    Turns out 1) is recursive and 2) is not. I therefore suggest a slight edit: for your "proof" substitute "proves" or "provable" as appropriate.

    I think you could expand your "Second Incompleteness Theorem" a bit. I found I could only understand it because I know the original. That is, just add a bit more descriptively to make it more explicitly clear. .

    Something I'm a bit fuzzy on: if you add G to S as new axiom, you obtain (say) S'. From S' a new G' can be generated, and so forth into the transfinite. I am not sure how to understand "adding G as an axiom."
  • Pippen
    80
    Hi Tim,

    I add "If S is consistent then G (is not provable)" to S as an axiom. It then becomes clear why we can't prove the consistency of S within S because it would lead to a proof of G which is impossible by the First Incompleteness Theorem.
  • tim wood
    9.3k
    I add "If S is consistent then G (is not provable)" to S as an axiom.Pippen
    Not for me to argue against. Let's give it a trial run. If S is consistent then G (is not provable). Do you see a bit of darkness and lightning on the horizon? The issue is not the consistency of S, but rather the provability of the consistency of S. I think that makes a difference. Because if (simply) S were consistent, then G would be provable - oops! But as an added axiom, the consistency of S (I'm guessing) becomes provable in S'. Leaving, one supposes, the consistency of S' to be provable in S''. The devil isn't just in the details; it is the details!

    But I think you've got it, imho.
  • A Seagull
    615
    My take on Gödel's incompleteness theorems is that they start with an initial assumption of completeness - ie that every statement that relates to a specific system S is either true or false. The proof (which incorporates basic maths) then leads to an inconsistency. Conclusion: no system that includes basic mathematics is complete.
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