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Godel’s Insight

Posted by Jerry on February 7, 2006

So, when Kurt Godel formulated the proof that there are true but unprovable statements, was he talking about axiomatic statements???
Say, for example, would the axiom that “existence exists” be the kind of statement that are unprovable but true?
Also, take another example: the validity of logic that cannot be proved, but it can be validated. Would that make the validity of the logical method an example of the unprovable but true? It seems to me that it would.

I’ve just started reading Incompleteness: The proof and paradox of Kurt Godel. I’m excited to read more!

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8 Responses to “Godel’s Insight”

  1. Mark G. said

    ” So, when Kurt Godel formulated the proof that there are true but unprovable statements, was he talking about axiomatic statements???”

    No. There is even an implication that if you were to take a statements that is true and couldn’t be proven and force them to be axioms, no matter how many times, then you would still have a system with true statements that couldn’t be proven.

    At least that’s what I remember from college so many years ago!

  2. Ergo Sum said

    “force them to be axioms, no matter how many times”

    How can one “force” a statement to be axioms? Aren’t axioms by their very definition, self-evidentiary and irreducible?

    How do you “force” the statement “Existence exists” to be an axiom… because any attempt to begin to “prove” its axiomatic nature will have to begin at a point assuming that it is NOT an axiom… which is impossible for these axiomatic statements. Any proof/attempt to validate or refute axioms must have to depend implicitly atleast on the axiom itself.

  3. Mark G. said

    How can one “force” a statement to be axioms?

    That may have been poor wording on my part. My experience with Godel is with mathematics. The math example that is most often cited is Number Theory which is repleat with statements that seem to be true, but can’t be proven and no counter examples can be found. If we take a statement such as “all even numbers can be written as the sum of two primes,” a very simple statement that mathematicians believe to be true but can’t prove, and then say O.K. let’s make that an axiom, then you end up with a new system of Number Theory with the added axioim. This new system in turn will have statements that can’t be proven true and no counter example can be found. Its been such a long time since I’ve read anything on Godel, but what I remember is the problem isn’t with mathematics itself. Statements in math are either true or false. Period. The problem is with the logic proving the statements.

    Thanks for bringing up the subject. It makes me want to go and brush up on Godel!

  4. Ergo Sum said

    Very interesting. I’m absolutely stupid when it comes to mathematics… so the only way I can work with Godel’s argument is by applying it to non-mathetical statements… like philosophical axioms.

    Yea… so far Godel comes across as an amazing, eccentric, genuis. I’m sure there’s something we’re missing… you are right that in Math things are either true or false (atleast that’s what I think, given my meager exposure to mathematics).

    So, given a statement, according to Godel, one could know it is true but not be able to prove it. Sounds to me like it’s not really a problem of logic, either.

    For example, “All bachelors are unmarried men”. Well, yeah… certainly, that’s true enough. But the statement is really self-referential… in that, it is a circular argument because it depends on the definition of the subject to conclude the predicate. Hence, it does not render itself to proof… but we still know its true.
    How is that a problem with logic? I see that as a matter of the identity of self-referential statements.
    And NO, I would not categorize that “bachelor” statement as an AXIOM. They are different things.

  5. You write, “So, when Kurt Godel formulated the proof that there are true but unprovable statements, was he talking about axiomatic statements???” No, he was talking about nonaxiomatic statements in a logical system (and it applies only to systems rich enough to include number theory). It’s an interesting fact about pure mathematics, but it doesn’t have much importance for human knowledge in general (contrary to the fashionable claims of many members of the intelligentsia), for which there are other criteria of truth than pure deduction from axioms.

  6. You write, “So, given a statement, according to Godel, one could know it is true but not be able to prove it. Sounds to me like it’s not really a problem of logic, either.” Uh, no, because it applies to deductive logical systems, not empirical sciences; it applies precisely to statements which you can’t know to be true unless they’re actually proved, such as statements of number theory. One could suspect a mathematical statement is true but not be able to prove it from a given set of axioms. More precisely, if you have a consistent set of axioms able to express arithmetic, there will be true statements that can’t be proved or disproved from those axioms. That means you can’t come up with an automatic, infallible algorithm or decision procedure that will produce all and only the true statements in arithmetic from a finite set of axioms (such as Peano’s axioms for defining the basic operations of arithmetic). (This is actually the gist of Church’s theorem, which is closely related to the work of Godel and which is of basic importance in theoretical computer science.) For example, it is undecidable in this sense whether a finite sequence of digits is truly random (in the strict sense of randomness used in computer science). You can’t prove that it’s truly random, but you can strongly suspect it.

  7. Ergo Sum said

    Here is the exact words of Godel’s argument as he presented it:

    “One can (assuming the consistency of classical mathematics) even give examples of propositions (and indeed of such a type as Goldbach and Fermat) which are really contextually true but unprovable in the formal system of classical mathematics.”

    Godel’s proof is known as the “Incompleteness” proof. Now, I suspect that you seem to have missed the point of his proof because you say his proof applies *only* to logical (mathematical) systems rich enough to include number theory.

    But, the point I think you’re missing is the epistemological point: How can Godel *PROVE* that there are some propositions that are both unprovable and true at the same time? To say that Godel *proves* that which turns out to be true but *unprovable* is not merely an issue of mathematics to grapple with, but a broader philosophical issue of the nature of proof and truth.

    My argument is that Godel’s insight is perfectly valid, but just like the propositions he describes as unprovable but true, I believe his theorem itself falls under that category of unprovable but true. I believe his theorem is *validated* by his mathematical logic, but not *proved*. There is a difference between showing validation and demonstrating proof.

    Furthermore, Godel is explicit in his qualification of “contextual” truths. That is infact also the Objectivist’s position on truth or certainty – that it can be arrived at only contextually.

  8. You write, “There is a difference between showing validation and demonstrating proof.” NO, NOT IN MATHEMATICS! As I pointed out in my recent comment about irrational numbers, mathematics proves theorems precisely by showing that they follow validly from a given set of consistent axioms and definitions; methodologically it is purely deductive, and in a deductive system validation IS proof. By the same token, Godel’s theorem is highly complex but valid for all mathematical systems rich enough to contain arithmetic, but to argue from that that it has any importance for other systems that do not rely exclusively on deduction for proof is invalid.

    You further write, “But, the point I think you’re missing is the epistemological point: How can Godel *PROVE* that there are some propositions that are both unprovable and true at the same time?” In short, by showing that if you start with a mathematical model of deductive logic that represents logical elements and functions as numerals (prime factors of an integer taken to represent the statement as a whole), you can model all forms of valid deduction in arithmetic; then he showed in essence that the statement “This statement is not decidable in this system” leads to a contradiction. It’s simply reductio ad absurdum at a very abstract level.

    “To say that Godel *proves* that which turns out to be true but *unprovable* is not merely an issue of mathematics to grapple with, but a broader philosophical issue of the nature of proof and truth.” Only if you assume that mathematical truth is just truth writ large, and mathematical proof is just like scientific proof. They’re not. You should read the following thread carefully:

    http://forums.4aynrandfans.com/index.php?showtopic=1732

    I’ll just quote Stephen Speicher’s comment from that thread, “Now, why should you care about all this? In normal life, not the least. As Goedel himself admitted, his work placed certain restrictions on the pure formalism of mathematics and had nothing to do with the ordinary reasoning power of the human mind.”

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