“I was simply trying to figure out that if so, what’s the “actual reality”?”
There is none, at least not in those terms. There is no “actual positional configuration space”, any more than there’s an “actual inertial reference frame” or “actual coordinate system”; they are all equivalent in the experimental world. Feel free to use whichever one you like.
“I’d thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable.”
The number of functions of the real line is actually strictly greater than beth-one uncountability (by Cantor’s theorem).
“I’d thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable.”
The category of Hilbert spaces includes spaces of both finite and infinite dimension, so it presumably includes both countable and uncountable infinities.
“I was simply trying to figure out that if so, what’s the “actual reality”?”
There is none, at least not in those terms. There is no “actual positional configuration space”, any more than there’s an “actual inertial reference frame” or “actual coordinate system”; they are all equivalent in the experimental world. Feel free to use whichever one you like.
“I’d thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable.”
The number of functions of the real line is actually strictly greater than beth-one uncountability (by Cantor’s theorem).
“I’d thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable.”
The category of Hilbert spaces includes spaces of both finite and infinite dimension, so it presumably includes both countable and uncountable infinities.