if there is no way to justify the premise that the world exists without resort to assumptions, then we're no better than the people who believe in God on faith.

Let us imagine two people. One believes "on faith" that (1) what their senses tell them has some correlation with how things really are, (2) their memory has some correlation with the past, and (3) their reasoning isn't completely random and broken. The other believes "on faith" those three things, and also that every statement in a certain collection of ancient documents is true, that a certain person who lived 2000 years ago was really a god in human form, that our true selves are immortal immaterial entities, and that after our deaths we will be judged and consigned to eternal bliss or eternal torment.

I'm quite happy saying that the first of those people is doing better than the second. S/he needs to assume far less; the things s/he assumes are more obviously true and more obviously unavoidable assumptions; there is less arbitrariness to them.

Both of them, indeed, fail if you judge them according to the following principle: "Everything you believe should be derived from absolutely incontestable axioms with which no one could possibly disagree". But why on earth should we do that?

"Self-evident" in the sense that they don't need any starting assumptions whatsoever.

Have you ever seen such a proposition? I don't think that I have. Not a single sage of recorded history has been able to come up with something whose self-evidence convinced everyone. And if someone is unconvinced, how shall you convince them, if it's "self-evident"?

The point I am making repeatedly because others don't seem to get it is that if there is no way to justify the premise that the world exists without resort to assumptions, then we're no better than the people who believe in God on faith.

What is this? If you have any unjustified belief, you are identical with someone who pays no heed to rationality at all?

And what does this have to do with probability in particular? You originally asked about probability, so I recommended works on the foundations. Even if none of them persuade you that they are a sound basis, at least you will be informed about the arguments and conceptual structures that people have created, at which point you may be able to productively search for something better.

But now you have broadened this to a requirement for a refutation of the Evil Demon/Matrix scenario. I see no possibility of any such refutation, because sufficient powers can always be attributed to the Demon/Skynet/Lizard Overlords/NSA to explain away any putative refutation. If there is a refutation, you will have to find it yourself.

I mentioned John C. Wright earlier, and there is more to say. He finds the ultimate foundation in the uncaused cause that is the Originator of all causation, the Good that needs no justification because it is the Originator of all that is good, proves their existence by the argument against infinite regress, and recognises them in the world as the Christian God, specifically as preached by the Roman Catholic Church. You could work out from that what his self-evident truths might be, for him to build these arguments on, but his actual self-evident truths are the religious visions that he had. He was never argued into any of this by the arguments that he presents (and neither am I, an atheist).

Self-evidence is a subjective property of a belief. The experience of self-evidence is the absence of experience of justification for the thing believed.