19 February 2010

Gödel's Incompleteness Theorems: Foreground

Enter Gödel, exit certainty


This post is part of a series on Gödel's incompleteness theorem. You may find all of them here:
  1. Gödel's incompleteness theorem: Background.

  2. Gödel's incompleteness theorem: Foreground.

  3. Gödel's incompleteness theorem: Underground.


Kurt Gödel accepted Hilbert's challenge. Sort of.

Gödel's first incompleteness theorem


If the system is consistent, it cannot be complete.


This means that if there are no contradictions in your system, the system cannot be complete. A contradiction occurs when a statement has no definite truth value. That is, when you cannot determine whether a statement is true or false. An example of such a statement in our ordinary system of language is the liar's paradox ("This sentence is false").

This sentence is false


If this is true, then the sentence is false. But if the sentence is false, it cannot be true. Now all the white rabbits go tumbling down the rabbit hole and Alice comes tumbling after into Blunderland.

For the record, it doesn't mean that if there are contradictions in your system that your system is complete. It works like this:
  • A cause leads to an effect.

  • We determine the cause, therefore we know that it would lead to the effect. We thus know the effect. This is called modus ponens.

  • We determine the effect, but we cannot determine the cause, all things being equal. To assert the cause from the effect is called the converse error.

This actually makes perfect sense. From modus ponens, we know that Chuck Norris causes broken legs without fail. We see Chuck Norris working on someone's legs. We can therefore be certain that Chuck Norris broke both their legs.

But the converse isn't true - we can't be certain that everyone with broken legs had a tussle with Chuck Norris (though I wouldn't take that bet) because this is the converse error.

Chuck Norris Wikipedia image

Chuck Norris proves modus ponens. And he doesn't even know it.

Similarly, from Gödel's first incompleteness theorem, we know that a system being consistent implies that it is incomplete (we have the cause, so we know the effect). But to claim that a system being incomplete implies that it is consistent is to commit the converse error (we have the effect, so we cannot determine the cause).

Many religious people state that the wonderful universe we live in was caused by god. Therefore, the wonderful universe we live in proves the existence of god. See the resemblance to the converse error?

Another fan favourite is irreducible complexity. Irreducible complexity - however defined - is an effect. An effect does not imply a cause, and to claim so is to commit the converse error, as we have seen.

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