Is it possible to make a knife/sword cartoonishly sharp?

[Solifugae]

Imagine you accidentally drop a kitchen knife and it cuts your fridge door in half. Is that actually even hypothetically possible, or completely barred by the law of the physics?
Also, what about the ridiculous trope popularized by kids during the 80's where katanas are able to cut through 10s of stacked bodies, and cut through houses etc etc etc. Obviously that's fictional as regards an actual katana, but is it possible to actually have those properties in a weapon. Course, that wouldn't be very practical, but that wouldn't stop someone doing it because it would be really really cool.

How sharp does physics actually allow things to be? I'm presuming the harder something it is, the sharper the edge it can hold, so if there's a limit to how hard materials can actually be, then that's also the limit to how fine a stable edge you can have.

 

[orienteer]

You are dealing with two properties here.

1) how strong can a material be? The strongest material known to man is diamonds.

2) What is the sharpest angle possible in nature? This goes back to Chemistry, and is based on the angles that the electron-nucleus-electron form, which in turn determines the shape of the covalent bonds to the atom next door.