Posted by: justpeachie | October 18, 2008

pants on fire

This sentence is false.

That’s a bold statement, really. Is it true or false?

If we give it a little bit of thought, we find that there is no answer: the sentence cannot be true, yet it cannot be false. If it is declared true, then it is false; if we call it false, then it is necessarily true. Therein lies the headache. Ultimately, what the Liar’s Paradox is conveying is not even its truth or falsehood, but the fact that it is unproveable.

That brings us to Kurt Godel, the Austrian man who (theoretically) turned the world of mathematics inside out back in the 1930s with his Incompleteness Theorems, the gist of which is that a theory T cannot be both complete and consistent within itself. He asserts:

Within any consistent, formal arithmetic theory, there still exist true statements which cannot be proven.

This is basically the super-abstract-higher-math version of a modified Liar’s Paradox. If Godel’s assertion is true, that means that the consistent theory is incomplete. If false, then the complete theory is inconsistent. In this way, Godel proves that that any consistent theory T is incomplete.

…and yet, contrary to what you’re currently thinking, it is a big deal! Keep in mind that Godel’s contemporaries (and even mathematicians now) strive for simplicity and conciseness. At the beginning of the 20th century, this “formalist system” was very trendy among hip mathematicians, who refined and re-refined their fields to just a few axioms. (I suppose there was really nothing else to do during the Great Depression.) S/he who could establish a formal theory on the most concise axiomatic foundations was Math Superman.

…then Kurt Godel invented Kryptonite. But that’s life: you lose some… and you lose some.

P.S. To learn about Godel’s sexual fetishes, click here.

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Responses

  1. Great post! To clarify slightly, what you call “Godel’s assertion” is a special statement (a statement that asserts its own unprovability, a la liar paradox) which he uses to prove the theorem.

    As you write, it is a big deal. Mathematicians were convinced that all true statements in mathematics could be proven from the axioms, and this was a source of comfort. Now we have to wonder if the theorem we are trying to prove can be proven at all…


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