Dedekind-Cantor Axiom: What's Needed & Why?

[Werg22]

I know that the Dedekind-Cantor axiom establishes an isomorphism between the points of any given (extended) Euclidean line. But why is the axiom needed anyway? Can't we define two binary operations on collinear points in Euclidean geometry such that the points of the line taken together with these two operations form a model of the real numbers?

 

[Take_it_Easy]

The axiom of Dedekind is an axiom regarding the EXISTENCE of the points on any real line.
All we know form previous algebraic axioms is that 0 is a real numer (neutral element for addition) and that 1 is a real number (neutral element for product).
It is IMPORTANT that 1 \not= 0 and this is in fact another axiom.
These axioms (together with other axioms on sum and products) lead to the existence of a countable multitude of numers in the real line. Namely \mathbb Q.
\mathbb Q itself satisfy every axioms except the Dedekind axiom.
So if you don't insert another axiom you are implicitly assuming that \mathbb Q is a satysfactory system for the real numbers.
That's why we require another axiom.

 

[Werg22]

Ah I see. Are the least upper bound axiom and the Dedekind-Axiom essentially equivalent?

 

[Take_it_Easy]

Ah I see. Are the least upper bound axiom and the Dedekind-Axiom essentially equivalent?

I am tempted to say "YES IT IS", but I don't know the least uper bound axiom.
What does it states?

 

[Werg22]

If S is a set of real numbers, k is called an upper bound and is a real number if for every s in S s <= k (I'm sure you know this). An upper bound of S, j, is called the least upper bound of S if j <= k for all k's. The axiom states that every set of real numbers possesses a least upper bound. Q fails to satisfy this axiom; suffice to construct a a series composed of rational numbers but converging to an irrational number.

 

[Dragonfall]

The Cantor-Dedekind axiom is not used in the construction of the real numbers. It's more like a metamathematical axiom which says that the "line" of geometry and the "real numbers" are pretty much the same thing. It's more along the lines of Cauchy's delta-epsilon definition of "continuity" and the Church-Turing thesis.

On the other hand, the least-upper-bound axiom is actually used (in standard analysis) to construct the reals from the rationals.