Existence of the Reals

We'll prove the following important theorem:

∃R

∃ an ordered field R which contains Q which satisfies the the axiom of completeness.

The problem was that Q couldn't guarantee the irrationals like 2, which if we try to define limits later would be a problem. The resolution here required the axiom of completeness, which was a definition needed to make R. Hence, it's not to be proved. But proving our theorem above will show that is indeed a valid theorem.

Different Ways of Doing This

While many mathematicians of the older times formulated various ways to define R, they all work! However, here we'll focus on one such construction by Dedekind.

We had:

  1. Dedekind cuts
  2. We then define R as the set of all cuts in Q. Even though we think of elements in R as numbers, we use the sets to create a correlation between the irrationals.
  3. We can go through all the properties of R as defined this way to verify it has the properties of a Field, and had Order. We define Addition on R (via Dedekind Cuts) and multiplication in a similar way.

References

  1. [[Abbott Real Analysis.pdf#page=297]]