>>12576776Lol, what do you call the symbol ? Since math is abstract, what about the set of all points in a line segment? What would you call the collection of all those points if not a set?
>there are no actually infinite collections of anything that we have access to and can manipulateI disagree. What about the set of all positions between two marks on a ruler? In quantum mechanics the state of being at each of those positions forms a Hilbert space such the set of the Hilbert space's spanning basis vectors is an infinite set. We can definitely access and manipulate these states. They've been doing it for about 100 years.
>how do we know that the theory is even consistent?You can do direct experiments to show that position eigenstates are accurately treated as orthogonal directions in a infinite dimensional Hilbert space. The basis vectors of any vector space can be assembled into a set. The set of the basis vectors in the state of positions between two marks on a ruler is an infinite set. To show that this is not some one-off coincidence, you can do the same thing with the spanning basis vectors of an infinite dimensional Hilbert space of momentum eigenstates. A Hilbert space is just a vector space and it is totally Cartesian in nature so we're not doing anything tricky between real analysis and quantum mechanics: we can easily test the validity of theories which use the idea of an infinite set.
How about the set of all energy eigenstates for an electron in a hydrogen atom? He we can verify the theory again for the case of a countably infinite set of basis vector in the Hilbert space as opposed to the above two examples regarding uncountably infinite sets of position and momentum eigenstates.
Fundamentally, not being able to disprove the theory doesn't prove that there are infinite sets. The thing that "proves" the existence of infinite sets, for the purposes of allowing an intuitive axiom, is that "unbounded" means "infinite."
