What do you call somebody who thinks they know everything?

None of them means "know-it-all". Perhaps "know-it-all" is the phrase you need?

 

Most of these words have echoes in pride - which doesn't necessarily mean false belief merely how someone behaves.

Arrogant - an exaggerated sense of who they are. Doesn't have to apply to knowledge though you can be arrogant about anything from looks to beekeeping ability
Does not have to be a knowledge of everything.

Conceited - this is being overly proud of yourself and is more about looking down on people. It doesn't actually have to be for any reason so doesn't work as the person wouldn't need to think they knew everything - just need you to think you are inferior in some way.

Egotistic/Egotistical - this means self absorbed and is more about how the person feels about themself and relates to you.

Scornful - this is just someone who doesn't approve of something you are doing and isn't very nice about it.

Vain - similar to conceited, more often used in terms of personal appearance but doesn't have to be.

Contemptuous/Disdainfu - see Scornful
l
Smug - See conceited.

A know-it-all, Smarty-pants, pedant, dogmatist, pedagogue etc may be all those things but they don't mean know-it-all.

Sophist can mean someone who puts forward a clever, reasoned but false arguement - however again that can be calculated rather than false belief.

 

I think, all in, I liked the suggestion of swat... not with Digtig's suggested emendation.. but with another.

 

*clears throat*

From the thesaurus...

Main Entry: know-it-all
Part of Speech: noun
Definition: one who claims full knowledge
Synonyms: braggart, brain, brain*, intellectual, smart aleck, smart aleck, smart-ass , smarty-pants, walking encyclopedia, windbag, wise guy, wiseacre

* = informal/non-formal usage

In other, related, entries, I also found "show-off" and "swellhead".

 

Not egotistical--egotistical just means someone is self-centered or thinks too highly of himself, with an "undue sense of self-importance."

 

There is no reason to give up on this site. I've read over several words the users have given, you have quite a few to chose from.