How to properly define completeness of a set

Question

A few weeks back I picked up a 1960 copy of General Theory of Functions and Integration by Taylor at a half price bookstore.

I started reading this and got up to the definition of completeness of an ordered field which they defined as follows (paraphrasing)

Divide your ordered field into an upper and lower partition $(L,R)$ via some inequality. An ordered field is complete iff for all possible partitions, either $L$ has a maximum or $R$ has a minimum

Now I was used to the definition

An ordered field is complete iff every bounded subset $S$ has a least upper bound and greatest lower bound.

Going off memory I might have phrased that wrong, but I know these two statements are equivalent. However my question is when proving completeness for the first time, what is the correct definition to use?

Answer 1

Of course your second definition is more useful in our daily work and gives quick results in situations coming up hundreds of times. But it involves "all subsets of ${\mathbb R}$", and this is indeed a large community of objects. For an axiom one would like simpler setups. That's what your first definition does; it just talks about splittings of ${\mathbb R}$ into a lower and an upper part: Any such splitting hits a unique number (which then can belong to the lower or the upper part).

Answer 2

I was used to the following definition:

An ordered field $R$ is complete² if every subset of $R$ that is bounded above has a least upper bound (in $R$).

This definition of completeness is (apparently) different from the one you were used:

An ordered field $R$ is complete¹ if every bounded subset of $R$ that has a least upper bound and a greatest lower bound (in $R$).

Now it is quite easy to prove that an ordered field is complete² if and only if it is complete¹ (see here, for instance). So, what is the definition of completeness? Both are equivalent, and there is no logical reason to prefer one or the other, since both of them denote the same object (the set of real numbers). In other words, the two definitions of completeness are extensionally identical.

Of course, the two definition of completeness are intensionally different, in the sens that they describe the same object in two (slightly) different ways, but none of them is "more correct" than the other, because both of them talk about the same object. Is there any difference if I define Rome as "the current capital of Italy" or as "the city of the Colosseum"?

Intensional differences can be relevant from a pedagogical point of view, because a way to describe an object might involve more technical notions with respect to another approach; or from a pragmatical point of view, because one of the ways to present the object might be more convenient (in the sense that it asks for less logical steps) to prove certain properties; but they are irrelevant from a logical point of view.

The difference between the definitions of complete² and complete¹ might seem ridiculous, but what I said is still valid if you use other definitions of completeness, such as the convergence of every Cauchy sequence, or the one you mentioned about partitions.