Are you sure you have two apples on your desk? What happens if you take a small bite? Is it still two apples? When will said apple "cease" to exist as an unit while you're eating it?
If the Old Greeks [1] haven't been able to solve this problem I'm not sure we'll be able to do better than them.
I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. At some point some will start disagreeing, but with two big full apples, all of them will agree.
Or is it more helpful if I say I have 10^27 atoms in my body? Quarks? Gluons? Strings from string theory? Still finite number, not "biteable", not infinite.
> I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition.
That’s useful, but very vague, and very movable criteria for “exists”. (Approx. Must be based in physical reality + enough people subjectively agree on it)
Do following finite numbers exist: -1, 0, 0.5, PI, 2^300 (more than particles in observable universe), sqrt(-1)? Do individual digits exist? If PI exist, how many digits does it have? Do models and algorithms in general exist? Do model existance depend on limits of your/somebodies capacity to understand them?
I would propose alternative, but useful way to look at this. “Two” and “infinity” are models. Both these models are useful, but “two” is just more common one. (Still, various infinities are useful for bunch of people)
Going by my (bad) analogy, in theory, yes, that is what that would mean.
More to the point, if we're not able to decide whenever it is exactly that an apple stops being an apple once we start eating it that could mean that the apple-ness of said apple has big chances of remaining intact (together with the underlying apple) irrespective of us eating it fully, and hence (theoretically) giving us an infinite supply of apples (or access to apple-ness, to be more exact).
But, going back to the Old Greeks, this is a very old question that we haven't been able to fully solve, I honestly think we'll never be able to solve it. Aristotle's focus on categories and especially his tertium non datur thing has allowed us to solve some practical problems (for example by allowing us to build a world that is based on techne, which among other things has allowed us to have this conversation here on the internet), but the bigger and more philosophical question of One vs. Infinity and everything in between remains un-solved.
Later edit: On Cantor vs. Aristotle, from here [1]
> Thus Cantor believed that Aristotle was quite mistaken in his analysis of the infinite, and that his authority was exceedingly detrimental
When it comes to the philosophy of mathematics and to philosophy itself I'm with Cantor on this. Could be that mainstream mathematics itself could benefit were we to ditch Aristotle and fully embrace Cantor, but I'm not a mathematician nor smart enough to say if that's in the realm of the possible.
If the Old Greeks [1] haven't been able to solve this problem I'm not sure we'll be able to do better than them.
[1] https://en.wikipedia.org/wiki/Parmenides