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Dredging up my old college math. A mathematician named Kantor proved that the number of rational numbers (fractions) of postive integers is the same as the numb

Dredging up my old college math. A mathematician named Kantor proved that the number of rational numbers (fractions) of postive integers is the same as the number of positive integers. His proof involves COUNTING the fractions. And that kind of infinity is called, well, countable or aleph sub(0). It's like O(n) in algorithms. And in that world O(polynomial) and O(n) are both countable. But our favorite transcendental numbers, you know them, pi, e, psi, that lot, are not part of that. Neither are multiples, or fractions, of those numbers. There's an uncountable infinity as well, holding them, and it's strictly larger than countable infinity. Maybe that what Walt Whitman was thinking when he wrote "I contain multitudes"? At any rate, possible physical distances are uncountable. Yup. There's more of them than there are of integers. And living things with brains have a (probably) countable number of neuronal interconnections, each of which depends on uncountable physical distances. (We know this in the computer industry: we have all sorts of hardware and software that quantizes the physical stuff going on in chips and conductors to extract bits -- to make the uncountable countable.) My question: is this digital infinity countably infinite? Or does it go beyond that? Do people who model -- information-theorically -- living brains and the minds they hold consider this issue? Does this countability matter to our understanding?

A_D_E_P_T 4y ago
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
    [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.
atleastoptimal 4y ago
I'd wager it's uncountably infinite. Here's my very vague justification. Something countably infinite proceeds towards infinity in one direction. Let's say we were at a store containing an infinite number of grocery items, there would be an infinite number of words signifying, so in a language which could only be the expression of listing items in that store, it would be countably infinite. The thing about real languages though is that there is an infinite number of possible interrelations between any two words based on context. This is similar to the uncountable infinite of the real numbers, in which any two rational numbers have an infinite number of real numbers between them.
- feoren 4y ago
  > Something countably infinite proceeds towards infinity in one direction. This is a mental trap when thinking about these things. In fact the rationals are also countably infinite (a/b, with a and b integers), as are the "complex integers", that is, the set (a + bi) with a and b integers, and i = sqrt(-1). The "2-dimensional" set of tuples (x, y) with x and y integers is countably infinite; in fact any n-dimensional set of tuples of integers is also, for finite n. Many more things are countable than most people realize. > The thing about real languages though is that there is an infinite number of possible interrelations between any two words based on context. The set of all possible books is countably infinite. Think about that for a bit. If your "context" can be encoded in any number of millions of books, encyclopedias, and dictionaries, then I can simply append that context onto whatever else I'm saying, and now I have my text plus all context. Even if you say "but books can't capture the subtle tonalities and facial expressions of human expression", you still have to realize that we have finite photorecptors in our eyes, and finite hairs in our ears, and finite neurons to process all that information. So the set of signals a human could ever possibly process as distinct must be (at most) countably infinite as well. You cannot get uncountably infinite language without infinitely large brains. I know we want to all feel like we're magic and special, and this mysterious uncountable infinity feels like it leaves lots of room for us to have magical consciousness and a soul and an afterlife and X-Men superpowers, but it just isn't there. It just doesn't work. Besides, don't underestimate the size of countable infinity. That's not exactly something to get claustrophobic about.
- 4y ago
feoren 4y ago
> A mathematician named Kantor Cantor. Georg Cantor. > And that kind of infinity is called, well, countable or aleph sub(0). It's like O(n) in algorithms. It's not really at all connected to O(n), and only tenuously connected to Big-O notation at all. Big-O notation works over integers or reals, or even some other (possibly finite) sets. It doesn't make sense to say O(n) is countable any more than it makes sense to say that the line "y = 2x + 7" is countable. What would that mean? Especially if x and y are real numbers? > our favorite transcendental numbers, you know them, pi, e, psi, that lot, are not part of that. Neither are multiples, or fractions, of those numbers. True for pi and e, but what is psi? Do you mean phi, the golden ratio? Or do you really mean psi, the sum of the reciprocals of the Fibonacci numbers (I had to look this one up)? The golden ratio (phi) is not transcendental: phi * (1 - sqrt(5)) is -2. It doesn't look like it's known whether psi is transcendental or not. > At any rate, possible physical distances are uncountable. There's no particular reason to believe this is true, and some reason to believe it's not. Look up the "Planck length"; below this length it's not clear whether the concept of "distance" is even meaningful. > And living things with brains have a (probably) countable number of neuronal interconnections ... Not just countable neuronal interconnections: literally finite. Neurons have finite size and your brain isn't infinitely large (sorry). > ... each of which depends on uncountable physical distances. Pseudoscientific mumbo jumbo. Not even wrong. Literal nonsense. > My question: is this digital infinity countably infinite? Or does it go beyond that? It is countably infinite by definition. It's isomorphic to the free monoid over the (finite) digits. > Does this countability matter to our understanding? No. Uncountability is a curious feature of our model of real numbers. All models are wrong, but some models are useful. There's no real evidence that the uncountability of reals is an actual useful feature of that model, and not just a curious edge-case artifact. Most likely there is no physical analogue to uncountably infinite sets (my opinion, obviously). Am I nitpicking you? Details matter. You seem pretty careless with your facts here, which is a great way to accidentally spread disinformation. Maybe try to be more careful in the future.
kmeisthax 4y ago
AFAIK pi, e, psi, etc are countable. I mean, you can count them: pi is 1, e is 2, psi 3, etc. If you invent a new transcendental real number that'll be 4. etc. Adding all integer multiples and fractions is also countable, etc. Uncountability starts at the uncomputable reals: these are numbers that only exist as infinite collections of digits and don't have a more concise definition. This includes things like Chaitin's Constant, which is the probability that a randomly-generated Turing Machine will halt. The Cantor diagonalization argument also worked with uncomputable reals - again, infinite sequences of digits. Computable reals are countable. Physical distances are not uncountable. At the very least we know space cannot be infinitely subdivided: resolving any distance shorter than a Planck length creates black holes. So that would limit us to countable infinities. The digital infinity is likely not even infinite; just "way too big a state space for humanity to ever feasibly exhaust absent concerted effort to do so". Though, if you want a story that plays around with this, check out Melancholy Elephants[0]. [0] http://www.spiderrobinson.com/melancholyelephants.html http://www.spiderrobinson.com/melancholyelephants.html

A_D_E_P_T 4y ago

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
  [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.

atleastoptimal 4y ago

I'd wager it's uncountably infinite. Here's my very vague justification. Something countably infinite proceeds towards infinity in one direction. Let's say we were at a store containing an infinite number of grocery items, there would be an infinite number of words signifying, so in a language which could only be the expression of listing items in that store, it would be countably infinite. The thing about real languages though is that there is an infinite number of possible interrelations between any two words based on context. This is similar to the uncountable infinite of the real numbers, in which any two rational numbers have an infinite number of real numbers between them.

feoren 4y ago
> Something countably infinite proceeds towards infinity in one direction. This is a mental trap when thinking about these things. In fact the rationals are also countably infinite (a/b, with a and b integers), as are the "complex integers", that is, the set (a + bi) with a and b integers, and i = sqrt(-1). The "2-dimensional" set of tuples (x, y) with x and y integers is countably infinite; in fact any n-dimensional set of tuples of integers is also, for finite n. Many more things are countable than most people realize. > The thing about real languages though is that there is an infinite number of possible interrelations between any two words based on context. The set of all possible books is countably infinite. Think about that for a bit. If your "context" can be encoded in any number of millions of books, encyclopedias, and dictionaries, then I can simply append that context onto whatever else I'm saying, and now I have my text plus all context. Even if you say "but books can't capture the subtle tonalities and facial expressions of human expression", you still have to realize that we have finite photorecptors in our eyes, and finite hairs in our ears, and finite neurons to process all that information. So the set of signals a human could ever possibly process as distinct must be (at most) countably infinite as well. You cannot get uncountably infinite language without infinitely large brains. I know we want to all feel like we're magic and special, and this mysterious uncountable infinity feels like it leaves lots of room for us to have magical consciousness and a soul and an afterlife and X-Men superpowers, but it just isn't there. It just doesn't work. Besides, don't underestimate the size of countable infinity. That's not exactly something to get claustrophobic about.
4y ago

feoren 4y ago

> A mathematician named Kantor Cantor. Georg Cantor. > And that kind of infinity is called, well, countable or aleph sub(0). It's like O(n) in algorithms. It's not really at all connected to O(n), and only tenuously connected to Big-O notation at all. Big-O notation works over integers or reals, or even some other (possibly finite) sets. It doesn't make sense to say O(n) is countable any more than it makes sense to say that the line "y = 2x + 7" is countable. What would that mean? Especially if x and y are real numbers? > our favorite transcendental numbers, you know them, pi, e, psi, that lot, are not part of that. Neither are multiples, or fractions, of those numbers. True for pi and e, but what is psi? Do you mean phi, the golden ratio? Or do you really mean psi, the sum of the reciprocals of the Fibonacci numbers (I had to look this one up)? The golden ratio (phi) is not transcendental: phi * (1 - sqrt(5)) is -2. It doesn't look like it's known whether psi is transcendental or not. > At any rate, possible physical distances are uncountable. There's no particular reason to believe this is true, and some reason to believe it's not. Look up the "Planck length"; below this length it's not clear whether the concept of "distance" is even meaningful. > And living things with brains have a (probably) countable number of neuronal interconnections ... Not just countable neuronal interconnections: literally finite. Neurons have finite size and your brain isn't infinitely large (sorry). > ... each of which depends on uncountable physical distances. Pseudoscientific mumbo jumbo. Not even wrong. Literal nonsense. > My question: is this digital infinity countably infinite? Or does it go beyond that? It is countably infinite by definition. It's isomorphic to the free monoid over the (finite) digits. > Does this countability matter to our understanding? No. Uncountability is a curious feature of our model of real numbers. All models are wrong, but some models are useful. There's no real evidence that the uncountability of reals is an actual useful feature of that model, and not just a curious edge-case artifact. Most likely there is no physical analogue to uncountably infinite sets (my opinion, obviously). Am I nitpicking you? Details matter. You seem pretty careless with your facts here, which is a great way to accidentally spread disinformation. Maybe try to be more careful in the future.

kmeisthax 4y ago

AFAIK pi, e, psi, etc are countable. I mean, you can count them: pi is 1, e is 2, psi 3, etc. If you invent a new transcendental real number that'll be 4. etc. Adding all integer multiples and fractions is also countable, etc. Uncountability starts at the uncomputable reals: these are numbers that only exist as infinite collections of digits and don't have a more concise definition. This includes things like Chaitin's Constant, which is the probability that a randomly-generated Turing Machine will halt. The Cantor diagonalization argument also worked with uncomputable reals - again, infinite sequences of digits. Computable reals are countable. Physical distances are not uncountable. At the very least we know space cannot be infinitely subdivided: resolving any distance shorter than a Planck length creates black holes. So that would limit us to countable infinities. The digital infinity is likely not even infinite; just "way too big a state space for humanity to ever feasibly exhaust absent concerted effort to do so". Though, if you want a story that plays around with this, check out Melancholy Elephants[0]. [0] http://www.spiderrobinson.com/melancholyelephants.html http://www.spiderrobinson.com/melancholyelephants.html