It is interesting that in almost all basic philosophical discussions you can hint to the absolute. In this case: Is there an absolute truth? Lets assume: "there is no absolute truth". It s obvious that i cannot say "there is no absolute truth" without contradicting myself, because the statement itself claims to be absolutely true. Thus, there is absolute truth.

"What if something would come up and show you the entire world is fake. Everything you believed to be true is shattered, broken down to binary code. Would that make the old world false?"
Indeed, in the case of many things you believed to be real/true, you will have to say that they have been unreal, an illusion, not true. But the old world itself would only be "false" concerning a certain attribute: physical reality. If you say "the Matrix World and all its entities are physically existing", this is false. Everything else, like, that you can actually eat bananas there, have feelings, etc. is true! The fact that those actions/feelings are illusions made by a program doesnt affect the fact, that you can take those actions/feelings there. Only the ontological status changes: all inside the Matrix is an illusion - but, as an illusion it is true. Inside the Matrix Cypher says: "I know that this steak isnt real" which means: Cypher knows that it is true, that this steak isnt real. So, to repeat, only certain aspects of the "old world" are false. And yes, if the "new world" would be a fake world again, we would have to reconsider what is real/true again. But we could e.g. argue, that certain aspects of reality, like the rules of logical thinking or mathematics are necessarily true in all possible worlds.

random thoughts: Many philsophers would say that 3-2=1 is always true, in all possible worlds.
Descartes says, my own subjectivity, the pure "i am thinking" is absolutely clear and true: Even if something/someone deceives him, this proves to him, that he has to be, because otherwise noone would be there to be deceived.

I really like this!