5 ms·

I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1

I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1 annex also use Dedekind's cuts.

fweimer 2y ago
It looks like Hardy used Dedekind cuts from starting with the second edition (1914), but not in the first edition (1908). What's the advantage of Dedekind cuts over say equivalence classes of Cauchy sequences of rational numbers? Particularly if you start out by introducing the integers and rational numbers as equivalence classes as well.
- zozbot234 2y ago
  Cauchy sequences can be made constructive (providing a nice foundation for numerical analysis); Dedekind cuts cannot.
- jostylr 2y ago
  The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. For example, a sequence of 0,0,0,...,0 where the number of dots is the count of all the atoms in the universe and then followed by the decimal approximations of pi: 3, 3.1, 3.14, 3.141, ... The key component is the error clause of getting close, but that can vary greatly from sequence to sequence as to when that happens. The cute idea of being able to look at a sequence and see roughly where it is converging just is not captured well in the reality of the equivalence classes. More or less, one can think of a Cauchy sequence of generating intervals that contain the real number, but it can be arbitrarily long before the sequence gets to "small" intervals. So comparing two Cauchy sequences could be quite difficult. Contrast that with the rational numbers where a/b ~ c/d if and only if ad = bc. This is a relatively simple thing to check if a, b, c, and d are comfortably within the realm of human computation. Dedekind cuts avoid this as there is just one object and it is assumed to be completed. This is unrealistic in general though the n-roots are wonderful examples to think it is all okay and explicit. But if one considers e, it becomes clear that one has to do an approximation to get bounds on what is in the lower cut. The (lower) Dedekind cut can be thought of as being the set of lower endpoints of intervals that contain the real number. My preference is to define real numbers as the set of inclusive rational intervals that contain the real number. That is a bit circular, of course, so one has to come up with properties that say when a set of intervals satisfies being a real number. The key property is based on the idea behind the intermediate value theorem, namely, given an interval containing the real number, any number in the interval divides the interval in two pieces, one which is in the set and the other is not (if the number chosen "is" the real number, then both pieces are in the set). There is a version of this idea which is theoretically complete and uses Dedekind cuts to establish its correctness[1] and there is a version of this idea which uses what I call oracles that gets into the practical messiness of not being able to fully present a real number in practice[2]. 1: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-dedekind-cuts.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl... 2: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-oracles.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
  - zozbot234 2y ago
    > The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. ... This can be addressed practically enough by introducing the notion of a 'modulus of convergence'.
  - red_trumpet 2y ago
    What you are referring to is also called the Principle of Nested Intervals: https://en.wikipedia.org/wiki/Nested_intervals#The_construction_of_the_real_numbers https://en.wikipedia.org/wiki/Nested_intervals#The_construct...
  - thaumasiotes 2y ago
    > The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. What's the misleading part of this supposed to be?
    - denotational 2y ago
      I’m not who you replied to, but: The equivalence classes of integers: pairs of naturals with (a, b) ~ (c, d) := (a + d) = (b + c). The equivalence classes of rationals: pairs of integers with (a, b) ~ (c, d) := ad = bc. It’s “easy” to tell whether two integers/rationals are equivalent, because the equivalence rule only requires you to determine whether one pair is a translation/multiple resp. of the other (proof is left to the reader). Cauchy sequences, on the other hand, require you to consider the limit of an infinite sequence; as the GP points out, two sequences with the same limit may differ by an arbitrarily large prefix, which makes them “hard” to compare. We can formalise this notion by pointing out that equality of integers and rationals is decidable, whereas equality of Cauchy reals is not. On the other hand, equality of Dedekind reals isn’t decidable either, so it’s not that Cauchy reals are necessarily easier than Dedekind reals, but more that they might lull one into a false sense of security because one might naively believe that it’s easy to tell if two sequences have the same limit.

I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1 annex also use Dedekind's cuts.

fweimer 2y ago

It looks like Hardy used Dedekind cuts from starting with the second edition (1914), but not in the first edition (1908). What's the advantage of Dedekind cuts over say equivalence classes of Cauchy sequences of rational numbers? Particularly if you start out by introducing the integers and rational numbers as equivalence classes as well.

zozbot234 2y ago
Cauchy sequences can be made constructive (providing a nice foundation for numerical analysis); Dedekind cuts cannot.
jostylr 2y ago
The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. For example, a sequence of 0,0,0,...,0 where the number of dots is the count of all the atoms in the universe and then followed by the decimal approximations of pi: 3, 3.1, 3.14, 3.141, ... The key component is the error clause of getting close, but that can vary greatly from sequence to sequence as to when that happens. The cute idea of being able to look at a sequence and see roughly where it is converging just is not captured well in the reality of the equivalence classes. More or less, one can think of a Cauchy sequence of generating intervals that contain the real number, but it can be arbitrarily long before the sequence gets to "small" intervals. So comparing two Cauchy sequences could be quite difficult. Contrast that with the rational numbers where a/b ~ c/d if and only if ad = bc. This is a relatively simple thing to check if a, b, c, and d are comfortably within the realm of human computation. Dedekind cuts avoid this as there is just one object and it is assumed to be completed. This is unrealistic in general though the n-roots are wonderful examples to think it is all okay and explicit. But if one considers e, it becomes clear that one has to do an approximation to get bounds on what is in the lower cut. The (lower) Dedekind cut can be thought of as being the set of lower endpoints of intervals that contain the real number. My preference is to define real numbers as the set of inclusive rational intervals that contain the real number. That is a bit circular, of course, so one has to come up with properties that say when a set of intervals satisfies being a real number. The key property is based on the idea behind the intermediate value theorem, namely, given an interval containing the real number, any number in the interval divides the interval in two pieces, one which is in the set and the other is not (if the number chosen "is" the real number, then both pieces are in the set). There is a version of this idea which is theoretically complete and uses Dedekind cuts to establish its correctness[1] and there is a version of this idea which uses what I call oracles that gets into the practical messiness of not being able to fully present a real number in practice[2]. 1: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-dedekind-cuts.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl... 2: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-oracles.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
- zozbot234 2y ago
  > The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. ... This can be addressed practically enough by introducing the notion of a 'modulus of convergence'.
- red_trumpet 2y ago
  What you are referring to is also called the Principle of Nested Intervals: https://en.wikipedia.org/wiki/Nested_intervals#The_construction_of_the_real_numbers https://en.wikipedia.org/wiki/Nested_intervals#The_construct...
- thaumasiotes 2y ago
  > The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. What's the misleading part of this supposed to be?
  - denotational 2y ago
    I’m not who you replied to, but: The equivalence classes of integers: pairs of naturals with (a, b) ~ (c, d) := (a + d) = (b + c). The equivalence classes of rationals: pairs of integers with (a, b) ~ (c, d) := ad = bc. It’s “easy” to tell whether two integers/rationals are equivalent, because the equivalence rule only requires you to determine whether one pair is a translation/multiple resp. of the other (proof is left to the reader). Cauchy sequences, on the other hand, require you to consider the limit of an infinite sequence; as the GP points out, two sequences with the same limit may differ by an arbitrarily large prefix, which makes them “hard” to compare. We can formalise this notion by pointing out that equality of integers and rationals is decidable, whereas equality of Cauchy reals is not. On the other hand, equality of Dedekind reals isn’t decidable either, so it’s not that Cauchy reals are necessarily easier than Dedekind reals, but more that they might lull one into a false sense of security because one might naively believe that it’s easy to tell if two sequences have the same limit.