What does Godël’s Incompleteness Theorem actually mean and imply? I just saw Ted-Ed’s video on this topic and didn’t fully understand what it means or what the implications of this are.

Question

What does Godël’s Incompleteness Theorem actually mean and imply? I just saw Ted-Ed’s video on this topic and didn’t fully understand what it means or what the implications of this are.

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Answer 1

Imagine that mathematical theorems are physical buildings. If a theorem is true, that means the building can be built and won’t just fall down.

Buildings are built with bricks, mortar, steel beams, etc. These are the building blocks. Math similarly has building blocks called axioms.

So say someone has said “I’m pretty sure we can build a building that looks like *this picture*”. People toil away until they figure out which building blocks to use and how, then they go and build it. Voila, they have just proved that building can be built (the theorem is proven).

But now imagine some builder comes along and shows that there must be some buildings that will not fall down, but cannot be built with any building blocks we have no matter how hard we try, and no matter what set of building blocks we use.

This is a nightmare for builders. We want to not only be able to build everything, we want to build it with as limited of a set of building blocks as possible. And we definitely don’t want perfectly good buildings to be unbuildable using our tools. But it turns out that no matter what, we can’t, and we just have to accept that we can’t build some buildings.

Edit: I’ll just add that what I described is called the *consistency* of math. Godel’s theorem actually comes in two parts, the other concerning the *completeness* of math.

Using the same analogy it would go something like this.

We *can* have limited systems which are consistent. We can have systems where we’re only concerned with brick buildings. In that system, we *can* build all possible “good” brick buildings. The obvious problem is that our system is incomplete. We can only build brick buildings, not every kind of building.

The full incompletes thereom basically says you can have one or the other, but not both. You can either be able to build all brick buildings and be limited in that way, or be able to build every kind of building, but not be able to build some of them. But you can’t have both consistency and completeness.