3 ms·

Physical distances are "uncountable" only if physical space is infinitely divisible. If there's no continuum, and if reality is granular -- even at a resolutio

Physical distances are "uncountable" only if physical space is infinitely divisible. If there's no continuum, and if reality is granular -- even at a resolution well below the Planck Length -- then all physical distances in space are countable. Digital infinity is by definition countable. There's no reason to assume that anything in our universe is actually uncountable -- as far as we know, it can all be simulated mathematically without invoking Cantor's hierarchies. This isn't necessarily a finitist position. It's just to say that the uncountable infinities don't necessarily interact with any known universe -- digital or otherwise.

emmelaich 4y ago
I suspect it's not infinitely divisible. My stupid argument is to ask whether you can be say pi metres away from something else. You'd think so because as you move somewhere between 3.142 and 3.143 metres away from something, you'd pass pi and therefore land right on it. But how do you find where to stop at this transcendental position? Having granular space would solve this because there would be no such position.
- A_D_E_P_T 4y ago
  Yeah, I agree. Here's another way to look at it: The Planck length is ~10^-35m. The diameter of the electron is not more than 10^-22m. It may be as small as 10^-88m. (The Planck length sets limits on our ability to measure at such resolutions, but doesn't imply that nothing smaller can exist.) If "bedrock reality" is granular, it may have an ultimate resolution of roughly 10^-90 to 10^-100m. This is small, but it's very, very far from infinitely small. Now imagine a circle. In a universe that is not infinitely divisible, there's no such thing as a perfect circle. A circle of any sort will always have "pixels" at, say, a resolution of 10^-95m. Thus we dispense with pi. And, in any case, you can't divide physical reality past that point -- so if you take a needle with a tip that measures 10^-95m, and you move that tip around a coordinate space, it will always be in a definite location that can be represented as some form of countable number.
- deleted 4y ago
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mxkopy 4y ago
The Planck length is often confused to mean the smallest discrete unit of space, but it's more like the size of the smallest 'window' through which information can travel without becoming a black hole. The endpoints of that 'window' can be at arbitrary coordinates, so in theory you can make arbitrarily precise measurements.

emmelaich 4y ago

I suspect it's not infinitely divisible. My stupid argument is to ask whether you can be say pi metres away from something else. You'd think so because as you move somewhere between 3.142 and 3.143 metres away from something, you'd pass pi and therefore land right on it. But how do you find where to stop at this transcendental position? Having granular space would solve this because there would be no such position.

A_D_E_P_T 4y ago
Yeah, I agree. Here's another way to look at it: The Planck length is ~10^-35m. The diameter of the electron is not more than 10^-22m. It may be as small as 10^-88m. (The Planck length sets limits on our ability to measure at such resolutions, but doesn't imply that nothing smaller can exist.) If "bedrock reality" is granular, it may have an ultimate resolution of roughly 10^-90 to 10^-100m. This is small, but it's very, very far from infinitely small. Now imagine a circle. In a universe that is not infinitely divisible, there's no such thing as a perfect circle. A circle of any sort will always have "pixels" at, say, a resolution of 10^-95m. Thus we dispense with pi. And, in any case, you can't divide physical reality past that point -- so if you take a needle with a tip that measures 10^-95m, and you move that tip around a coordinate space, it will always be in a definite location that can be represented as some form of countable number.
deleted 4y ago
[deleted]

mxkopy 4y ago

The Planck length is often confused to mean the smallest discrete unit of space, but it's more like the size of the smallest 'window' through which information can travel without becoming a black hole. The endpoints of that 'window' can be at arbitrary coordinates, so in theory you can make arbitrarily precise measurements.