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I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimag

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly paradoxical, and fascinatingly debatable in a "is a hot dog a sandwich" sort of way. But reframe it as a process (or as code), and all confusion disappears: for how many loops would you like to proceed? If you never stop, 0.99999... clearly approaches 1, without ever reaching it, and asking if they're the "same" is as academic as asking if the Ship of Theseus is the same ship, or if an electron is the same entity from one picosecond to the next.

ta1234567890 6y ago
> As a static number, the concept of infinity makes no sense; but once reimagined as a process Super insightful. That's the key right there. The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.
- thom 6y ago
  Yes, although for me this conception of infinity as a process also captures why there are probably no actual infinite things in the universe, only in silly games with numbers.
  - effie 6y ago
    Could you elaborate? Why infinite process cannot be actual thing in the universe? It's not like we know the start and the end date of the universe...
TuringTest 6y ago
I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach[1] showed me that there's another, more static logical interpretation of infinity which also comes handy. In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the generation process (i.e. it is higher than any item in the list, or smaller, or has happened after all of them)... This object doesn't really belong to the infinite generative sequence, it's an item outside all of them, and can't be reached by completing the sequence; it merely exist outside the process and happens to have the property of "dominating" all the items in it. You can assume the existence in the same way you assume the existence of a number which is the square root of -1, or how you define triangles whose angles add up to more or less than 180 degrees. If you do that, this object "at the infinite" can be formally defined and treated axiomatically to find out what mathematical properties it possesses. [1] https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
- Gehinnn 6y ago
  This thought can be made much more concrete when talking about sets. Clearly we can understand a collection of things as a set. Clearly numbers are things, so we can talk about the set of all numbers. But how many elements are in this set? A clever answer could be: It has as many elements as there are natural numbers! As subsets are also a thing, we could ask next how many subsets the set of all numbers has. A clever, but very problematic answer could be "as many as there are natural numbers!"
  - empath75 6y ago
    That’s actually not true, the power set of the naturals is uncountable.
    - Gehinnn 6y ago
      As I said, it would be a very problematic answer. But only if you properly define when two non-finite sets have the same size, you can lead this to a contradiction. Infinity can be understood intuitively, extending it to the cardinal numbers not.
RHSeeger 6y ago
> it could just as easily be reframed as 0.000...0001 = 0 But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.
- lukifer 6y ago
  My argument is that "noun" is a purely human abstraction, and that phenomena that act noun-like in nature are at best snapshots of iterative processes. Within the bounds of the noun abstraction, sure, I'll cede that point. But if one eschews that abstraction and looks at it purely as a process (I want to render 1/3 in decimal notation, then multiply that decimal notation by 3), there is always that niggling 0.000...1 remainder at every snapshot. The "never stopping" bit is what smuggles verbiness into the "0.999..." noun, while simultaneously pretending it's a static value.
  - RHSeeger 6y ago
    Regardless of whether or not they are human abstractions, a process (making cheese) and a noun (cheese) are two distinct things.
- deleted 6y ago
  [deleted]
- drran 6y ago
  0.000...1 can be written as 1/inf, which has sense in Surreal Numbers math.
  - nybble41 6y ago
    You could just as easily say that 0.000…54234 can be written as 1/∞. Surreal Numbers is a bit of a detour in this case. The premise of the idea that 1 - 0.999… could be written 0.000…1 is the mistaken concept that there is some point "after an infinite number of steps" where the expansion of 0.999… stops and you can leave the remaining 1. The expansion never stops and there is no final remainder. The result is 0.000…, which is more typically written as 0. Plain zero, not an infinitesimal. Personally I find the following algebraic proof to be the most approachable: x = 0.999… x = 0.9 + 0.0999… x = 0.9 + (0.999… ÷ 10) x = 0.9 + (x ÷ 10) x - 0.9 = x ÷ 10 10x - 9 = x 9x - 9 = 0 9x = 9 x = 1 0.999… = 1

ta1234567890 6y ago

> As a static number, the concept of infinity makes no sense; but once reimagined as a process Super insightful. That's the key right there. The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.

thom 6y ago
Yes, although for me this conception of infinity as a process also captures why there are probably no actual infinite things in the universe, only in silly games with numbers.
- effie 6y ago
  Could you elaborate? Why infinite process cannot be actual thing in the universe? It's not like we know the start and the end date of the universe...

TuringTest 6y ago

I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach[1] showed me that there's another, more static logical interpretation of infinity which also comes handy. In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the generation process (i.e. it is higher than any item in the list, or smaller, or has happened after all of them)... This object doesn't really belong to the infinite generative sequence, it's an item outside all of them, and can't be reached by completing the sequence; it merely exist outside the process and happens to have the property of "dominating" all the items in it. You can assume the existence in the same way you assume the existence of a number which is the square root of -1, or how you define triangles whose angles add up to more or less than 180 degrees. If you do that, this object "at the infinite" can be formally defined and treated axiomatically to find out what mathematical properties it possesses. [1] https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

Gehinnn 6y ago
This thought can be made much more concrete when talking about sets. Clearly we can understand a collection of things as a set. Clearly numbers are things, so we can talk about the set of all numbers. But how many elements are in this set? A clever answer could be: It has as many elements as there are natural numbers! As subsets are also a thing, we could ask next how many subsets the set of all numbers has. A clever, but very problematic answer could be "as many as there are natural numbers!"
- empath75 6y ago
  That’s actually not true, the power set of the naturals is uncountable.
  - Gehinnn 6y ago
    As I said, it would be a very problematic answer. But only if you properly define when two non-finite sets have the same size, you can lead this to a contradiction. Infinity can be understood intuitively, extending it to the cardinal numbers not.

RHSeeger 6y ago

> it could just as easily be reframed as 0.000...0001 = 0 But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.

lukifer 6y ago
My argument is that "noun" is a purely human abstraction, and that phenomena that act noun-like in nature are at best snapshots of iterative processes. Within the bounds of the noun abstraction, sure, I'll cede that point. But if one eschews that abstraction and looks at it purely as a process (I want to render 1/3 in decimal notation, then multiply that decimal notation by 3), there is always that niggling 0.000...1 remainder at every snapshot. The "never stopping" bit is what smuggles verbiness into the "0.999..." noun, while simultaneously pretending it's a static value.
- RHSeeger 6y ago
  Regardless of whether or not they are human abstractions, a process (making cheese) and a noun (cheese) are two distinct things.
deleted 6y ago
[deleted]
drran 6y ago
0.000...1 can be written as 1/inf, which has sense in Surreal Numbers math.
- nybble41 6y ago
  You could just as easily say that 0.000…54234 can be written as 1/∞. Surreal Numbers is a bit of a detour in this case. The premise of the idea that 1 - 0.999… could be written 0.000…1 is the mistaken concept that there is some point "after an infinite number of steps" where the expansion of 0.999… stops and you can leave the remaining 1. The expansion never stops and there is no final remainder. The result is 0.000…, which is more typically written as 0. Plain zero, not an infinitesimal. Personally I find the following algebraic proof to be the most approachable: x = 0.999… x = 0.9 + 0.0999… x = 0.9 + (0.999… ÷ 10) x = 0.9 + (x ÷ 10) x - 0.9 = x ÷ 10 10x - 9 = x 9x - 9 = 0 9x = 9 x = 1 0.999… = 1