20 February 2010

Gödel's Incompleteness Theorems: Underground

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.


Gödel's second incompleteness theorem


The consistency of the axioms cannot be proven within the system.

This means that you cannot prove meta-statements from within the system. A meta-statement here is defined as a statement about the system. This causes a strange loop.

M.C. Escher Drawing Hands

What's more is if you have a system with meta-statements that are consistent, the system is inconsistent.

The weird and wonderful metaphysical wine rack


It should be noted that Gödel's incompleteness theorems only apply when:
  • Your system is expressive enough to model arithmetic.

  • Your system can determine what is an axiom in the system and what is not. That is, your system has to be recursive.


Suggest then that we have a wine rack that can only take red wine. Like my wine rack. We have a cellar filled with wine - both red and white - and I'd like to fill my wine rack with nothing but the finest red wine. Only there is a problem.

Red, red wine

I'd like to know if my wine rack can fit all the red wine in this particular cellar. My wine rack knows to only accept red wine, so this makes my wine rack consistent. But my wine rack will only be complete if my wine rack contains all the red wine in the cellar. My wine rack has a counter that can add wine and subtract wine, as I put bottles inside and as I remove bottles respectively. This means my wine rack can model arithmetic. In addition, this wine rack is incredibly smart and can determine whether I put red wine or any other kind of wine in it. If you want to put white wine or fortified wine in it, the wine rack shoots you with lasers. In a sense, the wine rack is recursive because it knows when a wine is red (and thus fit for the rack, or an axiom of the system), and when it is another (and thus unfit for the rack, or not an axiom of the system).

What Gödel's incompleteness theorems teaches us about metaphysical wine racks


From Gödel's first incompleteness theorem, we know that if a system is consistent, it cannot be complete. Thus we know that if my wine rack consistently accepts only red wine until it is full, it cannot be said to contain all the red wine in the cellar.

This means that when I pick a red wine with deep colour, rich palate, exquisite nose, tantalising body and of magnificent development, open it, let it breathe and take a sip, I have no way of proving whether it is from the cellar in general or from my incredibly fancy metaphysical wine rack. Especially not after a couple of glasses.

It does not mean that this red wine is either from the cellar or from my wine rack. Knowing me, I probably hid it in my coat. It also does not make the wine inside my wine rack or from the cellar any less red. Similarly, proven statements of determined truth value do not become untrue just because there are other statements in a system that are true but unprovable to be true. Likewise, it does not suddenly make any statement of indeterminable truth value necessarily true.

This is a grave mistake made by religious people who claim a syllogism as follows:
  • From Gödel's first incompleteness theorem, we know that if a formal system is incomplete, then statements exist within the system which can never be proven true or false within the system.

  • We cannot prove that God exists from within our system.

  • Therefore, God exists.


Firstly, religion is not a formal system. Like the Law of Attraction, it's not advanced enough to model arithmetic, so Gödel doesn't apply. Secondly, we also know that religious systems are inconsistent, even though some claim to be both complete and consistent. If Gödel did apply, we already know a system that can claim itself consistent is not. Lastly, even if we assume that a religious system is both formal and incomplete and consistent, the only conclusion is that statements exist of which we cannot determine the truth value. If 'God exists' is such a statement, then our conclusion is that 'God exists' is an indeterminable statement in our system. But we knew this already, so even if Gödel's theorems did apply to religions, they're not really advancing the debate.

Ford says Ford makes the best cars, so Ford makes the best cars


Clearly, this doesn't help us at all. It's like my metaphysical wine rack claiming that it contains all the red wine in our cellar. How does it know? It can only tally the wine coming into the rack or going out, but it hasn't counted all the wine in the cellar. If we want to determine whether all the red wine in the cellar is contained in the wine rack, we have to step outside the wine rack and tally the wine in the cellar. But if we climb out of the wine rack, the wine rack is no longer complete because it couldn't determine whether it contained all the wine in the cellar by itself.

Suggested reading on Gödel's Incompleteness Theorems


If you are interested, follow these helpful hints:

1 comment:

Ian said...

your wine rack scared me stone cold sober into drinking only red wine from now on

Creative Commons License