A major concern in epistemology is just what we can know about existential claims, that is, claims about the existence (or lack of) something.
Suppose for example that I assert the following:
The cake is a lie. No physical cake exists.
This is a negative existential claim, meaning that I am making a claim that some thing (in this case, cake) does not exist physically. Whether the "concept" of a cake exists or not in some metaphysical way is irrelevant; I am only concerned with the existence of real, delicious cake. While this is a simple claim, a huge question in epistemology is whether I can prove this or not.
There are a number of ways I could go about trying to prove my statement. I could start looking around my room, under tables, in drawers, anywhere I could imagine in search of cake. I could recruit a team of volunteers with the promise of cake and set them off in search of said pastry to see if there is any to be found. I could make some sort of cake radar and plant gigantic antennas everywhere to investigate the outer reaches of space for the existence of cakes (see also SETI).
However, through any of this, does contemporary philosophy hold that I can ever prove that the cake is a lie?
As an aside, what about a restricted claim? What if I instead assert that:
No physical cake exists in this room.
Is there a consensus on whether the positional restriction changes the provability of the claim? What does the major literature have to say about this?