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.
It is easy if you know the limits; if you don't, it's still true that two sequences {r_n}, {s_n} have the same limit if and only if the limit of the difference sequence {r_n - s_n} is zero, which conveniently enough is an integer and can't mess up our attempt to define the reals without invoking the reals.
That won't help you much if you don't know what you're working with, but the same is true of rationals.
I'm missing something as to this:
> equality of Dedekind reals isn’t decidable either
Two Dedekind reals (A, B) and (A', B') are equal if and only if they have identical representations. [Which is to say, A = A' and B = B'.] This is about as simple as equality gets, and is the normal rule of equality for ordered pairs. Can you elaborate on how you're thinking about decidability?
> Two Dedekind reals (A, B) and (A', B') are equal if and only if they have identical representations. […] Can you elaborate on how you're thinking about decidability?
Direct:
Make one of the sets uncomputable, at which point the equality of the sets cannot be decided. This happens when the real defined by the Dedekind cut is itself uncomputable. BB(764) is an integer (!) that I know is uncomputable off the top of my head. The same idea (defining an object in terms of some halting property) is used in the next proof.
Via undecidability of Cauchy reals:
Equality of Cauchy reals is also undecidable. The proof is by negation: consider a procedure that decides whether a real is equal to zero; consider a
sequence (a_n) with a_n = 1 if Turing machine A halts within n steps on all inputs, 0 otherwise; this is clearly Cauchy, but if we can decide whether it’s equal to 0, then we can decide HALT.
Cauchy reals and Dedekind reals are isomorphic, so equality of Dedekinds must also be undecidable.
Hopefully those two sketches show what I mean by decidable; caveat that I’m not infallible and haven’t been in academia for a while, so some/all of this may be wrong!
> BB(764) is an integer (!) that I know is uncomputable
I meant BB(748) apparently.
To elaborate on this point a bit, I specifically mean uncomputable in ZFC. There may be other foundations in which it is computable, but we can just find another n for which BB(n) is uncomputable in that framework since BB is an uncomputable function.
Your method for deciding whether two rationals are or aren't equal relies on having representations of those rationals. If you don't have those, it doesn't matter that there's an efficient test of equality when you do.
But you're arguing that equality of Dedekind reals is undecidable based on a problem that occurs when you define a particular "Dedekind real" only by reference to some property that it has. If you had a representation of the values as Dedekind reals, it would be trivial to determine whether they were or weren't equal. You're holding them to a different standard than you're using for the integers and rationals. Why?
Let's decide a question about the integers. Is BB(800) equal to BB(801)?
The important point for me is that the equivalence for Cauchy sequences are part of the definition of real numbers as Cauchy sequences. This ought to imply that one has to be able to decide equivalence of two sequences for the definition to make sense. For Dedekind cuts, the crucial aspect is being able to define the set and that is something that can be called into question. But if that is done, it is just a computational question in comparing two Dedekind cuts, not a definitional one.
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.