Re: Godel's Incompleteness and Nonmonotonic Logic

From: Aatu Koskensilta (aatu.koskensilta_at_xortec.fi)
Date: 08/28/04

  • Next message: Herman Jurjus: "Re: Godel's Incompleteness and Nonmonotonic Logic" 
    Date: Sat, 28 Aug 2004 12:35:39 +0300

    Xee wrote an amazingly confused post in an amusingly condescending tone,
    in which we find jewels like

    > The miracle of this is twofold: any
    > formula in the predicate calculus corresponds to a propositional sentence,
    > and every propositional sentence corresponds to a formula of the predicate
    > calculus. These theorems are known, respectively, as the Soundness and
    > Completeness Theorems of First Order Logic. Soundness, that every formula
    > describes a sentence, gets its name from the idea that our formulae don't
    > talk ***, they are always either true or false, never both nor neither,
    > guaranteed. Completeness, that every sentence can be described with some
    > formula, gets its name from the simple fact that there are no sentences we
    > can't describe, we're not missing any.

    Xee, may I ask you where you get these ideas? I particularly like the
    bit about completeness meaning that "every sentence can be described
    with some formula".

    > The Incompleteness Theorem arises in
    > mathematical logic whe one considers axioms by which arithmetic can be
    > derived. One such system, which Godel used, is known as Peano Arithmetic.
    > He showed that it is possible to say the following in Peano Arithmetic: All
    > sentences of the form (x) are false. This means that the very sentence
    > itself is false, since (x) has no qualifiers (i.e. any sentence is of the
    > form (x) because (x) can be anything). This is not a result in first order
    > logic!

    It's not a result of any logic I know of. You can't express arithmetical
    falsity in the language of arithmetic.

    > First order logic (propositional logic and predicate caluculs,
    > both) is Sound and Complete, no doubt about it. Godel's result is one in a
    > more general field of ...

    Perhaps you could have benefitted from reading the other posts in this
    thread where this confusion is - once again - corrected. 

    -- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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