>Reals as Cauchy sequences with N given as a function of epsilon (epsilon rational)
Advantages:
- Can add, subtract two numbers.
- Can multiply and divide two numbers.
- Can form exp(x) and many other operations involving infinite series.
- Possible to check that x!=y when two numbers x,y are NOT equal.
- From a Cauchy sequence x_n can obtain the limit x (provided the sequence has N attached as a function of epsilon)
Disadvantages:
- No way to check whether two numbers are equal (reduces to the Halting problem).

>Reals as Cauchy sequences WITHOUT N given as a function of epsilon (epsilon rational)
Advantages
- Can add, subtract, multiply, divide (provided we know the denominator is nonzero).
Disadvantages:
- Cannot form exp(x) nor most other operations involving infinite series.
- Impossible to detect when x!=y NOR when x=y. I.e. essentially no way to compare two real numbers.
- Cannot get the limit of a Cauchy sequence of reals.
>Reals as Dedekind cuts
Advantages:
- Possible to detect when two numbers x,y are unequal.
Disadvantages:
- No way to detect when two numbers x,y are equal.
- Cannot add, subtract, multiply, divide.
- Cannot take the limit of a Cauchy sequence.
- Cannot form exp(x) nor most other operations involving infinite series.

What am I missing something here? Are Cauchy sequences with N given the clear winner among the implementations of reals?